Logarithm - Wikipedia

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 2.1 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Logarithmic_identities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Logarithmic_identities"> <div class="vector-toc-text"> 3 Logarithmic identities </div> </a> <button aria-controls="toc-Logarithmic_identities-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Logarithmic identities subsection </button> <ul id="toc-Logarithmic_identities-sublist" class="vector-toc-list"> <li id="toc-Product,_quotient,_power,_and_root" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Product,_quotient,_power,_and_root"> <div class="vector-toc-text"> 3.1 Product, quotient, power, and root </div> </a> <ul id="toc-Product,_quotient,_power,_and_root-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Change_of_base" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Change_of_base"> <div class="vector-toc-text"> 3.2 Change of base </div> </a> <ul id="toc-Change_of_base-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Particular_bases" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Particular_bases"> <div class="vector-toc-text"> 4 Particular bases </div> </a> <ul id="toc-Particular_bases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 5 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logarithm_tables,_slide_rules,_and_historical_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Logarithm_tables,_slide_rules,_and_historical_applications"> <div class="vector-toc-text"> 6 Logarithm tables, slide rules, and historical applications </div> </a> <button aria-controls="toc-Logarithm_tables,_slide_rules,_and_historical_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Logarithm tables, slide rules, and historical applications subsection </button> <ul id="toc-Logarithm_tables,_slide_rules,_and_historical_applications-sublist" class="vector-toc-list"> <li id="toc-Log_tables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Log_tables"> <div class="vector-toc-text"> 6.1 Log tables </div> </a> <ul id="toc-Log_tables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computations"> <div class="vector-toc-text"> 6.2 Computations </div> </a> <ul id="toc-Computations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Slide_rules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Slide_rules"> <div class="vector-toc-text"> 6.3 Slide rules </div> </a> <ul id="toc-Slide_rules-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analytic_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Analytic_properties"> <div class="vector-toc-text"> 7 Analytic properties </div> </a> <button aria-controls="toc-Analytic_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Analytic properties subsection </button> <ul id="toc-Analytic_properties-sublist" class="vector-toc-list"> <li id="toc-Existence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Existence"> <div class="vector-toc-text"> 7.1 Existence </div> </a> <ul id="toc-Existence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characterization_by_the_product_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characterization_by_the_product_formula"> <div class="vector-toc-text"> 7.2 Characterization by the product formula </div> </a> <ul id="toc-Characterization_by_the_product_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graph_of_the_logarithm_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graph_of_the_logarithm_function"> <div class="vector-toc-text"> 7.3 Graph of the logarithm function </div> </a> <ul id="toc-Graph_of_the_logarithm_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivative_and_antiderivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivative_and_antiderivative"> <div class="vector-toc-text"> 7.4 Derivative and antiderivative </div> </a> <ul id="toc-Derivative_and_antiderivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_representation_of_the_natural_logarithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_representation_of_the_natural_logarithm"> <div class="vector-toc-text"> 7.5 Integral representation of the natural logarithm </div> </a> <ul id="toc-Integral_representation_of_the_natural_logarithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transcendence_of_the_logarithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transcendence_of_the_logarithm"> <div class="vector-toc-text"> 7.6 Transcendence of the logarithm </div> </a> <ul id="toc-Transcendence_of_the_logarithm-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Calculation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculation"> <div class="vector-toc-text"> 8 Calculation </div> </a> <button aria-controls="toc-Calculation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Calculation subsection </button> <ul id="toc-Calculation-sublist" class="vector-toc-list"> <li id="toc-Power_series" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power_series"> <div class="vector-toc-text"> 8.1 Power series </div> </a> <ul id="toc-Power_series-sublist" class="vector-toc-list"> <li id="toc-Taylor_series" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Taylor_series"> <div class="vector-toc-text"> 8.1.1 Taylor series </div> </a> <ul id="toc-Taylor_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverse_hyperbolic_tangent" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Inverse_hyperbolic_tangent"> <div class="vector-toc-text"> 8.1.2 Inverse hyperbolic tangent </div> </a> <ul id="toc-Inverse_hyperbolic_tangent-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Arithmetic–geometric_mean_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic–geometric_mean_approximation"> <div class="vector-toc-text"> 8.2 Arithmetic–geometric mean approximation </div> </a> <ul id="toc-Arithmetic–geometric_mean_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Feynman's_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Feynman's_algorithm"> <div class="vector-toc-text"> 8.3 Feynman's algorithm </div> </a> <ul id="toc-Feynman's_algorithm-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 9 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Logarithmic_scale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithmic_scale"> <div class="vector-toc-text"> 9.1 Logarithmic scale </div> </a> <ul id="toc-Logarithmic_scale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Psychology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Psychology"> <div class="vector-toc-text"> 9.2 Psychology </div> </a> <ul id="toc-Psychology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_theory_and_statistics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_theory_and_statistics"> <div class="vector-toc-text"> 9.3 Probability theory and statistics </div> </a> <ul id="toc-Probability_theory_and_statistics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_complexity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computational_complexity"> <div class="vector-toc-text"> 9.4 Computational complexity </div> </a> <ul id="toc-Computational_complexity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entropy_and_chaos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entropy_and_chaos"> <div class="vector-toc-text"> 9.5 Entropy and chaos </div> </a> <ul id="toc-Entropy_and_chaos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fractals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fractals"> <div class="vector-toc-text"> 9.6 Fractals </div> </a> <ul id="toc-Fractals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Music" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Music"> <div class="vector-toc-text"> 9.7 Music </div> </a> <ul id="toc-Music-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Number_theory"> <div class="vector-toc-text"> 9.8 Number theory </div> </a> <ul id="toc-Number_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 10 Generalizations </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations subsection </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Complex_logarithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_logarithm"> <div class="vector-toc-text"> 10.1 Complex logarithm </div> </a> <ul id="toc-Complex_logarithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverses_of_other_exponential_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverses_of_other_exponential_functions"> <div class="vector-toc-text"> 10.2 Inverses of other exponential functions </div> </a> <ul id="toc-Inverses_of_other_exponential_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_concepts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_concepts"> <div class="vector-toc-text"> 10.3 Related concepts </div> </a> <ul id="toc-Related_concepts-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 11 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 12 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 13 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Logarithm</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 109 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-109" aria-hidden="true" > 109 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Logaritme" title="Logaritme – Afrikaans" lang="af" hreflang="af" data-title="Logaritme" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Logarithmus" title="Logarithmus – Alemannic" lang="gsw" hreflang="gsw" data-title="Logarithmus" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%8E%E1%8C%8B%E1%88%AA%E1%8B%9D%E1%88%9D" title="ሎጋሪዝም – Amharic" lang="am" hreflang="am" data-title="ሎጋሪዝም" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D9%88%D8%BA%D8%A7%D8%B1%D9%8A%D8%AA%D9%85" title="لوغاريتم – Arabic" lang="ar" hreflang="ar" data-title="لوغاريتم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Aragonese" lang="an" hreflang="an" data-title="Logaritmo" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%98%E0%A6%BE%E0%A6%A4%E0%A6%BE%E0%A6%82%E0%A6%95" title="ঘাতাংক – Assamese" lang="as" hreflang="as" data-title="ঘাতাংক" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Logaritmu" title="Logaritmu – Asturian" lang="ast" hreflang="ast" data-title="Logaritmu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Loqarifm" title="Loqarifm – Azerbaijani" lang="az" hreflang="az" data-title="Loqarifm" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B2%E0%A6%97%E0%A6%BE%E0%A6%B0%E0%A6%BF%E0%A6%A6%E0%A6%AE" title="লগারিদম – Bangla" lang="bn" hreflang="bn" data-title="লগারিদম" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Logaritma" title="Logaritma – Banjar" lang="bjn" hreflang="bjn" data-title="Logaritma" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target">Banjar</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/T%C3%B9i-s%C3%B2%CD%98" title="Tùi-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Tùi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – Bashkir" lang="ba" hreflang="ba" data-title="Логарифм" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D0%B0%D1%80%D1%8B%D1%84%D0%BC" title="Лагарыфм – Belarusian" lang="be" hreflang="be" data-title="Лагарыфм" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%8F%D0%B3%D0%B0%D1%80%D1%8B%D1%82%D0%BC" title="Лягарытм – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лягарытм" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Logaritmo" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D1%8A%D0%BC" title="Логаритъм – Bulgarian" lang="bg" hreflang="bg" data-title="Логаритъм" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Logaritam" title="Logaritam – Bosnian" lang="bs" hreflang="bs" data-title="Logaritam" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Logaritm" title="Logaritm – Breton" lang="br" hreflang="br" data-title="Logaritm" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Логарифм" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Logaritme" title="Logaritme – Catalan" lang="ca" hreflang="ca" data-title="Logaritme" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – Chuvash" lang="cv" hreflang="cv" data-title="Логарифм" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Logaritmus" title="Logaritmus – Czech" lang="cs" hreflang="cs" data-title="Logaritmus" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Daraunene" title="Daraunene – Shona" lang="sn" hreflang="sn" data-title="Daraunene" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target">ChiShona</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Logarithm" title="Logarithm – Welsh" lang="cy" hreflang="cy" data-title="Logarithm" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Logaritme" title="Logaritme – Danish" lang="da" hreflang="da" data-title="Logaritme" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%84%D9%88%DA%AD%D8%A7%D8%B1%D9%8A%D8%AA%D9%85" title="لوڭاريتم – Moroccan Arabic" lang="ary" hreflang="ary" data-title="لوڭاريتم" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Logarithmus" title="Logarithmus – German" lang="de" hreflang="de" data-title="Logarithmus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Logaritm" title="Logaritm – Estonian" lang="et" hreflang="et" data-title="Logaritm" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%AC%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82" title="Λογάριθμος – Greek" lang="el" hreflang="el" data-title="Λογάριθμος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Logar%C3%ACtem" title="Logarìtem – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Logarìtem" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Spanish" lang="es" hreflang="es" data-title="Logaritmo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Esperanto" lang="eo" hreflang="eo" data-title="Logaritmo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Logaritmu" title="Logaritmu – Extremaduran" lang="ext" hreflang="ext" data-title="Logaritmu" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target">Estremeñu</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Basque" lang="eu" hreflang="eu" data-title="Logaritmo" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%84%DA%AF%D8%A7%D8%B1%DB%8C%D8%AA%D9%85" title="لگاریتم – Persian" lang="fa" hreflang="fa" data-title="لگاریتم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Logarithm" title="Logarithm – Fiji Hindi" lang="hif" hreflang="hif" data-title="Logarithm" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Logaritma" title="Logaritma – Faroese" lang="fo" hreflang="fo" data-title="Logaritma" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Logarithme" title="Logarithme – French" lang="fr" hreflang="fr" data-title="Logarithme" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Logartam" title="Logartam – Irish" lang="ga" hreflang="ga" data-title="Logartam" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Galician" lang="gl" hreflang="gl" data-title="Logaritmo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%B0%8D%E6%95%B8" title="對數 – Gan" lang="gan" hreflang="gan" data-title="對數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%9C%EA%B7%B8_(%EC%88%98%ED%95%99)" title="로그 (수학) – Korean" lang="ko" hreflang="ko" data-title="로그 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BC%D5%B8%D5%A3%D5%A1%D6%80%D5%AB%D5%A9%D5%B4" title="Լոգարիթմ – Armenian" lang="hy" hreflang="hy" data-title="Լոգարիթմ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B2%E0%A4%98%E0%A5%81%E0%A4%97%E0%A4%A3%E0%A4%95" title="लघुगणक – Hindi" lang="hi" hreflang="hi" data-title="लघुगणक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Logaritam" title="Logaritam – Croatian" lang="hr" hreflang="hr" data-title="Logaritam" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Ido" lang="io" hreflang="io" data-title="Logaritmo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logaritma" title="Logaritma – Indonesian" lang="id" hreflang="id" data-title="Logaritma" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Logarithmo" title="Logarithmo – Interlingua" lang="ia" hreflang="ia" data-title="Logarithmo" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Logri" title="Logri – Icelandic" lang="is" hreflang="is" data-title="Logri" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Italian" lang="it" hreflang="it" data-title="Logaritmo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9C%D7%95%D7%92%D7%A8%D7%99%D7%AA%D7%9D" title="לוגריתם – Hebrew" lang="he" hreflang="he" data-title="לוגריתם" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9A%E1%83%9D%E1%83%92%E1%83%90%E1%83%A0%E1%83%98%E1%83%97%E1%83%9B%E1%83%98" title="ლოგარითმი – Georgian" lang="ka" hreflang="ka" data-title="ლოგარითმი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – Kazakh" lang="kk" hreflang="kk" data-title="Логарифм" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Logi" title="Logi – Swahili" lang="sw" hreflang="sw" data-title="Logi" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Logaritm" title="Logaritm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Logaritm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logarithmus" title="Logarithmus – Latin" lang="la" hreflang="la" data-title="Logarithmus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Logaritms" title="Logaritms – Latvian" lang="lv" hreflang="lv" data-title="Logaritms" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Logaritmas" title="Logaritmas – Lithuanian" lang="lt" hreflang="lt" data-title="Logaritmas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Logaritmo" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Logaritm" title="Logaritm – Lombard" lang="lmo" hreflang="lmo" data-title="Logaritm" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://hu.wikipedia.org/wiki/Logaritmus" title="Logaritmus – Hungarian" lang="hu" hreflang="hu" data-title="Logaritmus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://mk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Логаритам – Macedonian" lang="mk" hreflang="mk" data-title="Логаритам" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Anisa" title="Anisa – Malagasy" lang="mg" hreflang="mg" data-title="Anisa" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B2%E0%B5%8B%E0%B4%97%E0%B4%B0%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ലോഗരിതം – Malayalam" lang="ml" hreflang="ml" data-title="ലോഗരിതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A5%85%E0%A4%B0%E0%A4%BF%E0%A4%A6%E0%A4%AE" title="लॉगॅरिदम – Marathi" lang="mr" hreflang="mr" data-title="लॉगॅरिदम" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logaritma" title="Logaritma – Malay" lang="ms" hreflang="ms" data-title="Logaritma" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9C%E1%80%B1%E1%80%AC%E1%80%B7%E1%80%82%E1%80%9B%E1%80%85%E1%80%BA%E1%80%9E%E1%80%99%E1%80%BA" title="လော့ဂရစ်သမ် – Burmese" lang="my" hreflang="my" data-title="လော့ဂရစ်သမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Logaritme" title="Logaritme – Dutch" lang="nl" hreflang="nl" data-title="Logaritme" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AF%BE%E6%95%B0" title="対数 – Japanese" lang="ja" hreflang="ja" data-title="対数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Logaritme" title="Logaritme – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Logaritme" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Logaritme" title="Logaritme – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Logaritme" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Logaritme" title="Logaritme – Occitan" lang="oc" hreflang="oc" data-title="Logaritme" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Loogarizimii" title="Loogarizimii – Oromo" lang="om" hreflang="om" data-title="Loogarizimii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Logarifm" title="Logarifm – Uzbek" lang="uz" hreflang="uz" data-title="Logarifm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A8%98%E0%A9%82%E0%A8%97%E0%A8%A3%E0%A8%95" title="ਲਘੂਗਣਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਲਘੂਗਣਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%84%D8%A7%DA%AF%D8%B1%D8%AA%DA%BE%D9%85" title="لاگرتھم – Western Punjabi" lang="pnb" hreflang="pnb" data-title="لاگرتھم" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Lagaridim" title="Lagaridim – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Lagaridim" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Logarithmus" title="Logarithmus – Low German" lang="nds" hreflang="nds" data-title="Logarithmus" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logarytm" title="Logarytm – Polish" lang="pl" hreflang="pl" data-title="Logarytm" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://pt.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Portuguese" lang="pt" hreflang="pt" data-title="Logaritmo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Logaritm" title="Logaritm – Romanian" lang="ro" hreflang="ro" data-title="Logaritm" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – Russian" lang="ru" hreflang="ru" data-title="Логарифм" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – Yakut" lang="sah" hreflang="sah" data-title="Логарифм" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logaritmet" title="Logaritmet – Albanian" lang="sq" hreflang="sq" data-title="Logaritmet" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Logaritmu" title="Logaritmu – Sicilian" lang="scn" hreflang="scn" data-title="Logaritmu" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BD%E0%B6%9D%E0%B7%94_%E0%B6%9C%E0%B6%AB%E0%B6%9A" title="ලඝු ගණක – Sinhala" lang="si" hreflang="si" data-title="ලඝු ගණක" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Logarithm" title="Logarithm – Simple English" lang="en-simple" hreflang="en-simple" data-title="Logarithm" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li 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Click here for more information."><img alt="Featured article" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/20px-Cscr-featured.svg.png" decoding="async" width="20" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/30px-Cscr-featured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/40px-Cscr-featured.svg.png 2x" data-file-width="466" data-file-height="443" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Base_of_a_logarithm&redirect=no" class="mw-redirect" title="Base of a logarithm">Base of a logarithm</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical function, inverse of an exponential function</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Logarithm_plots.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Logarithm_plots.png/300px-Logarithm_plots.png" decoding="async" width="300" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Logarithm_plots.png/450px-Logarithm_plots.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Logarithm_plots.png/600px-Logarithm_plots.png 2x" data-file-width="1706" data-file-height="1294" /></a><figcaption>Plots of logarithm functions, with three commonly used bases. The special points logb b = 1 are indicated by dotted lines, and all curves intersect in logb 1 = 0.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" 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.navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><td class="sidebar-above" style="background:#efefef;"> <a href="/wiki/Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">Arithmetic operations</a><div class="navbar plainlinks hlist navbar-mini" style="float:right"><ul><li class="nv-view"><a href="/wiki/Template:Arithmetic_operations" title="Template:Arithmetic operations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Arithmetic_operations" title="Template talk:Arithmetic operations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Arithmetic_operations" title="Special:EditPage/Template:Arithmetic operations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr><tr><td class="sidebar-content" style="font-size:130%;"> <table class="infobox-subbox infobox-3cols-child infobox-table"><tbody><tr><th colspan="4" class="infobox-header"><a href="/wiki/Addition" title="Addition">Addition</a> (+)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>summand</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>summand</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>augend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>addend</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea99a27b5a763ef48889c450ac8157083ea97118" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.217ex; height:9.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\text{sum}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sum</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{sum}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8609baca9fdbc4c529f5894884a08122d695dad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.931ex; height:1.343ex;" alt="{\displaystyle \scriptstyle {\text{sum}}}"></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Subtraction" title="Subtraction">Subtraction</a> (−)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>term</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>minuend</mtext> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>subtrahend</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2780b756445a5f8f95b16c33e3b924f976958ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.356ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\text{difference}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>difference</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{difference}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac22c4e24eef2036cff5bfea924cc0dbb30c5d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.857ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{difference}}}"></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Multiplication" title="Multiplication">Multiplication</a> (×)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>factor</mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>factor</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>multiplier</mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>multiplicand</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f7b476e32221c7b05d356289c8085aef54059b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.176ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <a href="/wiki/Product_(mathematics)" title="Product (mathematics)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\text{product}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>product</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{product}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c8b7509b8be1043622cb7b1b9a36ca8bfc2616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.578ex; height:1.843ex;" alt="{\displaystyle \scriptstyle {\text{product}}}"></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Division_(mathematics)" title="Division (mathematics)">Division</a> (÷)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="0.83em 0.4em" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>dividend</mtext> </mrow> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>divisor</mtext> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>numerator</mtext> </mrow> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>denominator</mtext> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5d22ff59234f0d437be740306e8dd905991e1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:14.15ex; height:8.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <a href="/wiki/Quotient" title="Quotient"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>fraction</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>quotient</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ratio</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2359c3ca6e50e7ae8065baa710440b3c79895023" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.197ex; height:7.176ex;" alt="{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}"></a></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Exponentiation" title="Exponentiation">Exponentiation</a> (^)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>exponent</mtext> </mrow> </msup> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>power</mtext> </mrow> </msup> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecb107371002b62a60fcbd13e742f4d81f872b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.618ex; height:4.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\text{power}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>power</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{power}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0d0a9fbffb659c0055d5ee6fde3f7f28e96f45c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.297ex; height:1.509ex;" alt="{\displaystyle \scriptstyle {\text{power}}}"></td></tr><tr><th colspan="4" class="infobox-header"><a href="/wiki/Nth_root" title="Nth root">nth root</a> (√)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>radicand</mtext> </mrow> </mstyle> <mrow class="MJX-TeXAtom-ORD"> <mtext>degree</mtext> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5582d567e7e7fbcdb728291770905e09beb0ea18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.422ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\text{root}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>root</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{root}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a015c1122190da3f1f1732d88b8bb03a8d7eb91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.928ex; height:1.676ex;" alt="{\displaystyle \scriptstyle {\text{root}}}"></td></tr><tr><th colspan="4" class="infobox-header"><a class="mw-selflink selflink">Logarithm</a> (log)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>base</mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>anti-logarithm</mtext> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2435266fcae4aa91d3d70a74bb91b5b35ef52edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.454ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}"></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\text{logarithm}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>logarithm</mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{logarithm}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5d50baa86b950ff6d15760b7a38df1f8d8c868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.948ex; height:2.009ex;" alt="{\displaystyle \scriptstyle {\text{logarithm}}}"></td></tr></tbody></table></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Arithmetic_operations" title="Template:Arithmetic operations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Arithmetic_operations" title="Template talk:Arithmetic operations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Arithmetic_operations" title="Special:EditPage/Template:Arithmetic operations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the logarithm to base b is the inverse function of <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> with base b. That means that the logarithm of a number x to the base b is the <a href="/wiki/Exponent" class="mw-redirect" title="Exponent">exponent</a> to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"> of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x. When the base is clear from the context or is irrelevant it is sometimes written log x. The logarithm base 10 is called the decimal or <a href="/wiki/Common_logarithm" title="Common logarithm">common logarithm</a> and is commonly used in science and engineering. The <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> has the number <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e ≈ 2.718</a> as its base; its use is widespread in mathematics and <a href="/wiki/Physics" title="Physics">physics</a> because of its very simple <a href="/wiki/Derivative" title="Derivative">derivative</a>. The <a href="/wiki/Binary_logarithm" title="Binary logarithm">binary logarithm</a> uses base 2 and is frequently used in <a href="/wiki/Computer_science" title="Computer science">computer science</a>. Logarithms were introduced by <a href="/wiki/John_Napier" title="John Napier">John Napier</a> in 1614 as a means of simplifying calculations.<a href="#cite_note-1">[1]</a> They were rapidly adopted by <a href="/wiki/Navigator" title="Navigator">navigators</a>, scientists, engineers, <a href="/wiki/Surveying" title="Surveying">surveyors</a>, and others to perform high-accuracy computations more easily. Using <a href="/wiki/Logarithm_table" class="mw-redirect" title="Logarithm table">logarithm tables</a>, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a> is the <a href="/wiki/Summation" title="Summation">sum</a> of the logarithms of the factors: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72599165912508b07108f2a840898022ed126148" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.868ex; height:2.843ex;" alt="{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,}"> provided that b, x and y are all positive and b ≠ 1. The <a href="/wiki/Slide_rule" title="Slide rule">slide rule</a>, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, who connected them to the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> in the 18th century, and who also introduced the letter e as the base of natural logarithms.<a href="#cite_note-2">[2]</a> <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">Logarithmic scales</a> reduce wide-ranging quantities to smaller scopes. For example, the <a href="/wiki/Decibel" title="Decibel">decibel</a> (dB) is a <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">unit</a> used to express <a href="/wiki/Level_(logarithmic_quantity)" title="Level (logarithmic quantity)">ratio as logarithms</a>, mostly for signal power and amplitude (of which <a href="/wiki/Sound_pressure" title="Sound pressure">sound pressure</a> is a common example). In chemistry, <a href="/wiki/PH" title="PH">pH</a> is a logarithmic measure for the <a href="/wiki/Acid" title="Acid">acidity</a> of an <a href="/wiki/Aqueous_solution" title="Aqueous solution">aqueous solution</a>. Logarithms are commonplace in scientific <a href="/wiki/Formula" title="Formula">formulae</a>, and in measurements of the <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity of algorithms</a> and of geometric objects called <a href="/wiki/Fractal" title="Fractal">fractals</a>. They help to describe <a href="/wiki/Frequency" title="Frequency">frequency</a> ratios of <a href="/wiki/Interval_(music)" title="Interval (music)">musical intervals</a>, appear in formulas counting <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> or <a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">approximating</a> <a href="/wiki/Factorial" title="Factorial">factorials</a>, inform some models in <a href="/wiki/Psychophysics" title="Psychophysics">psychophysics</a>, and can aid in <a href="/wiki/Forensic_accounting" title="Forensic accounting">forensic accounting</a>. The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithm</a> is the multi-valued <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> of the complex exponential function. Similarly, the <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a> is the multi-valued inverse of the exponential function in finite groups; it has uses in <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=1" title="Edit section: Motivation">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Binary_logarithm_plot_with_grid.png" class="mw-file-description"><img alt="Graph showing a logarithmic curve, crossing the x-axis at x= 1 and approaching minus infinity along the y-axis." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Binary_logarithm_plot_with_grid.png/300px-Binary_logarithm_plot_with_grid.png" decoding="async" width="300" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Binary_logarithm_plot_with_grid.png/450px-Binary_logarithm_plot_with_grid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Binary_logarithm_plot_with_grid.png/600px-Binary_logarithm_plot_with_grid.png 2x" data-file-width="1704" data-file-height="1292" /></a><figcaption>The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the logarithm base 2 crosses the <a href="/wiki/X_axis" class="mw-redirect" title="X axis">x-axis</a> at x = 1 and passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log2(8) = 3 and 23 = 8. The graph gets arbitrarily close to the y-axis, but <a href="/wiki/Asymptotic" class="mw-redirect" title="Asymptotic">does not meet it</a>.</figcaption></figure> <a href="/wiki/Addition" title="Addition">Addition</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> are three of the most fundamental arithmetic operations. The inverse of addition is <a href="/wiki/Subtraction" title="Subtraction">subtraction</a>, and the inverse of multiplication is <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>. Similarly, a logarithm is the inverse operation of <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>. Exponentiation is when a number b, the base, is raised to a certain power y, the exponent, to give a value x; this is denoted <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{y}=x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{y}=x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d862a261be92079096455ce1af882eb1c15f99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.122ex; height:2.343ex;" alt="{\displaystyle b^{y}=x.}"> For example, raising 2 to the power of 3 gives 8: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{3}=8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}=8.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3f620a5eabedc8d4791da4f800c50e570212a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.124ex; height:2.676ex;" alt="{\displaystyle 2^{3}=8.}"> The logarithm of base b is the inverse operation, that provides the output y from the input x. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\log _{b}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\log _{b}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181577dca8c4b040c22f3a8a794a772496c5ba67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.88ex; height:2.676ex;" alt="{\displaystyle y=\log _{b}x}"> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=b^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=b^{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed854c1c6aed9b31c9d1140d89cacb98c5229dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.475ex; height:2.343ex;" alt="{\displaystyle x=b^{y}}"> if b is a positive <a href="/wiki/Real_number" title="Real number">real number</a>. (If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.) One of the main historical motivations of introducing logarithms is the formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72599165912508b07108f2a840898022ed126148" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.868ex; height:2.843ex;" alt="{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y,}"> by which <a href="/wiki/Logarithm_table" class="mw-redirect" title="Logarithm table">tables of logarithms</a> allow multiplication and division to be reduced to addition and subtraction, a great aid to calculations before the invention of computers. <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=2" title="Edit section: Definition">edit</a>]</div> Given a positive <a href="/wiki/Real_number" title="Real number">real number</a> b such that b ≠ 1, the logarithm of a positive real number x with respect to base b<a href="#cite_note-3">[nb 1]</a> is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the unique real number y such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{y}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{y}=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b51b8dcca3e64714e2f3aba12bb8c68812f77cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.475ex; height:2.343ex;" alt="{\displaystyle b^{y}=x}">.<a href="#cite_note-4">[3]</a> The logarithm is denoted "logb x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x"). An equivalent and more succinct definition is that the function logb is the <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> to the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto b^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto b^{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/873987f9618fe2c30ce4e72cbd1a967ff759c1d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.114ex; height:2.343ex;" alt="{\displaystyle x\mapsto b^{x}}">. <div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=3" title="Edit section: Examples">edit</a>]</div> <ul><li>log2 16 = 4, since 24 = 2 × 2 × 2 × 2 = 16.</li> <li>Logarithms can also be negative: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{2}\!{\frac {1}{2}}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{2}\!{\frac {1}{2}}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8b94168610cb7ed4c84db6c07c952c86b090a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.753ex; height:3.509ex;" alt="{\textstyle \log _{2}\!{\frac {1}{2}}=-1}"> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7133bdd52feadc28a28fe4080716ebd22c813d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.487ex; height:4.009ex;" alt="{\textstyle 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}"></li> <li>log10 150 is approximately 2.176, which lies between 2 and 3, just as 150 lies between 102 = 100 and 103 = 1000.</li> <li>For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1, respectively.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Logarithmic_identities">Logarithmic identities</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=4" title="Edit section: Logarithmic identities">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_logarithmic_identities" title="List of logarithmic identities">List of logarithmic identities</a></div> Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.<a href="#cite_note-5">[4]</a> <div class="mw-heading mw-heading3"><h3 id="Product,_quotient,_power,_and_root">Product, quotient, power, and root</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=5" title="Edit section: Product, quotient, power, and root">edit</a>]</div> The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=b^{\,\log _{b}x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=b^{\,\log _{b}x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f8b602ea0c566a881f48b98516fcd1535adfc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.211ex; height:2.676ex;" alt="{\displaystyle x=b^{\,\log _{b}x}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=b^{\,\log _{b}y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=b^{\,\log _{b}y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45059aa4f1349f9b5fae86c0d31015b3f55e434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.913ex; height:3.009ex;" alt="{\displaystyle y=b^{\,\log _{b}y}}"> in the left hand sides. <table class="wikitable plainrowheaders"> <caption>Product, quotient, power, and root identities of logarithms </caption> <tbody><tr> <th scope="col">Identity </th> <th scope="col">Formula </th> <th scope="col">Example </th></tr> <tr> <th scope="row">Product </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{b}(xy)=\log _{b}x+\log _{b}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{b}(xy)=\log _{b}x+\log _{b}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4f0c2d6cea945fa78f11152fe8127a0649d65a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.221ex; height:2.843ex;" alt="{\textstyle \log _{b}(xy)=\log _{b}x+\log _{b}y}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{3}243=\log _{3}(9\cdot 27)=\log _{3}9+\log _{3}27=2+3=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mn>243</mn> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>9</mn> <mo>⋅</mo> <mn>27</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mn>9</mn> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mn>27</mn> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{3}243=\log _{3}(9\cdot 27)=\log _{3}9+\log _{3}27=2+3=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5edf603416a8561c82c90c121aef2db207d385fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.778ex; height:2.843ex;" alt="{\textstyle \log _{3}243=\log _{3}(9\cdot 27)=\log _{3}9+\log _{3}27=2+3=5}"> </td></tr> <tr> <th scope="row">Quotient </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a87dae414cbad13958321eb275688c81afc52c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.703ex; height:3.343ex;" alt="{\textstyle \log _{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{2}16=\log _{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>16</mn> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>64</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>64</mn> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>4</mn> <mo>=</mo> <mn>6</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{2}16=\log _{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb8136de3357cf09002709aa2c608431aa819ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:47.12ex; height:3.676ex;" alt="{\textstyle \log _{2}16=\log _{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}"> </td></tr> <tr> <th scope="row">Power </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{b}\left(x^{p}\right)=p\log _{b}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>p</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{b}\left(x^{p}\right)=p\log _{b}x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c0ee92ecd37daeb1e8e8375a09fc073f61b529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.389ex; height:2.843ex;" alt="{\textstyle \log _{b}\left(x^{p}\right)=p\log _{b}x}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log _{2}2=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>64</mn> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>6</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mn>2</mn> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log _{2}2=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9999f9b43799c609893876c509df0d2c8f6c752f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.693ex; height:3.343ex;" alt="{\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log _{2}2=6}"> </td></tr> <tr> <th scope="row">Root </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68ca3b6cc8ff1c0192fb0e9206d32b14aec60e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.662ex; height:4.176ex;" alt="{\textstyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1000</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>1000</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c3e999db5ec9dbc719770862d56466dcf2b4421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.676ex; height:3.509ex;" alt="{\textstyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Change_of_base">Change of base</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=6" title="Edit section: Change of base">edit</a>]</div> The logarithm logb x can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:<a href="#cite_note-6">[nb 2]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6575863d29b357a7ba1ce0a4c1bb93a53b19e671" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.985ex; height:6.176ex;" alt="{\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.}"> Typical <a href="/wiki/Scientific_calculators" class="mw-redirect" title="Scientific calculators">scientific calculators</a> calculate the logarithms to bases 10 and <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a>.<a href="#cite_note-7">[5]</a> Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b69f8613162e2ae7c46c99bc72402644e16ef317" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.394ex; height:6.176ex;" alt="{\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.}"> Given a number x and its logarithm y = logb x to an unknown base b, the base is given by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=x^{\frac {1}{y}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=x^{\frac {1}{y}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3fd9660ce8c1b5ba4ef5d2f1f08a8ed5dde5a08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.808ex; height:3.843ex;" alt="{\displaystyle b=x^{\frac {1}{y}},}"> which can be seen from taking the defining equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=b^{\,\log _{b}x}=b^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=b^{\,\log _{b}x}=b^{y}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2311f2504e15b12a0209f669134985b6b7deac7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.356ex; height:2.676ex;" alt="{\displaystyle x=b^{\,\log _{b}x}=b^{y}}"> to the power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{y}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88ffe0bbb43942183114c150f227a11184b4c11e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.305ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{y}}.}"> <div class="mw-heading mw-heading2"><h2 id="Particular_bases">Particular bases</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=7" title="Edit section: Particular bases">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Log4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Log4.svg/260px-Log4.svg.png" decoding="async" width="260" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Log4.svg/390px-Log4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Log4.svg/520px-Log4.svg.png 2x" data-file-width="575" data-file-height="500" /></a><figcaption>Overlaid graphs of the logarithm for bases <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠ 1 / 2 ⁠, 2, and e</figcaption></figure> Among all choices for the base, three are particularly common. These are b = 10, b = <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a> (the <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> mathematical constant <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a> ≈ 2.71828183 ), and b = 2 (the <a href="/wiki/Binary_logarithm" title="Binary logarithm">binary logarithm</a>). In <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the logarithm base e is widespread because of analytical properties explained below. On the other hand, base 10 logarithms (the <a href="/wiki/Common_logarithm" title="Common logarithm">common logarithm</a>) are easy to use for manual calculations in the <a href="/wiki/Decimal" title="Decimal">decimal</a> number system:<a href="#cite_note-8">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mspace width="thinmathspace" /> <mn>10</mn> <mspace width="thinmathspace" /> <mi>x</mi> <mspace width="thinmathspace" /> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>=</mo> <mspace width="thickmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>10</mn> <mtext> </mtext> <mo>+</mo> <mspace width="thickmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>1</mn> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa604c3444753377b366fd0e105138d408338f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.011ex; height:2.843ex;" alt="{\displaystyle \log _{10}\,(\,10\,x\,)\ =\;\log _{10}10\ +\;\log _{10}x\ =\ 1\,+\,\log _{10}x\,.}"> Thus, log10 (x) is related to the number of <a href="/wiki/Decimal_digit" class="mw-redirect" title="Decimal digit">decimal digits</a> of a positive integer x: The number of digits is the smallest <a href="/wiki/Integer" title="Integer">integer</a> strictly bigger than log10 (x) .<a href="#cite_note-9">[7]</a> For example, log10(5986) is approximately 3.78 . The <a href="/wiki/Floor_and_ceiling_functions" title="Floor and ceiling functions">next integer above</a> it is 4, which is the number of digits of 5986. Both the natural logarithm and the binary logarithm are used in <a href="/wiki/Information_theory" title="Information theory">information theory</a>, corresponding to the use of <a href="/wiki/Nat_(unit)" title="Nat (unit)">nats</a> or <a href="/wiki/Bit" title="Bit">bits</a> as the fundamental units of information, respectively.<a href="#cite_note-10">[8]</a> Binary logarithms are also used in <a href="/wiki/Computer_science" title="Computer science">computer science</a>, where the <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary system</a> is ubiquitous; in <a href="/wiki/Music_theory" title="Music theory">music theory</a>, where a pitch ratio of two (the <a href="/wiki/Octave" title="Octave">octave</a>) is ubiquitous and the number of <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a> between any two pitches is a scaled version of the binary logarithm, or log 2 times 1200, of the pitch ratio (that is, 100 cents per <a href="/wiki/Semitone" title="Semitone">semitone</a> in <a href="/wiki/12_tone_equal_temperament" class="mw-redirect" title="12 tone equal temperament">conventional equal temperament</a>), or equivalently the log base 21/1200  ; and in <a href="/wiki/Photography" title="Photography">photography</a> rescaled base 2 logarithms are used to measure <a href="/wiki/Exposure_value" title="Exposure value">exposure values</a>, <a href="/wiki/Luminance" title="Luminance">light levels</a>, <a href="/wiki/Exposure_time" class="mw-redirect" title="Exposure time">exposure times</a>, lens <a href="/wiki/Aperture" title="Aperture">apertures</a>, and <a href="/wiki/Film_speed" title="Film speed">film speeds</a> in "stops".<a href="#cite_note-11">[9]</a> The abbreviation log x is often used when the intended base can be inferred based on the context or discipline, or when the base is indeterminate or immaterial. Common logarithms (base 10), historically used in logarithm tables and slide rules, are a basic tool for measurement and computation in many areas of science and engineering; in these contexts log x still often means the base ten logarithm.<a href="#cite_note-12">[10]</a> In mathematics log x usually means to the natural logarithm (base e).<a href="#cite_note-13">[11]</a><a href="#cite_note-14">[12]</a> In computer science and information theory, log often refers to binary logarithms (base 2). The following table lists common notations for logarithms to these bases. The "ISO notation" column lists designations suggested by the <a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">International Organization for Standardization</a>.<a href="#cite_note-15">[13]</a> <table class="wikitable" style="text-align:center;"> <tbody><tr> <th scope="col">Base b </th> <th scope="col">Name for logb x </th> <th scope="col">ISO notation </th> <th scope="col">Other notations </th></tr> <tr> <th scope="row">2 </th> <td><a href="/wiki/Binary_logarithm" title="Binary logarithm">binary logarithm</a> </td> <td>lb x <a href="#cite_note-gullberg-16">[14]</a> </td> <td>ld x, log x, lg x,<a href="#cite_note-17">[15]</a> log2 x </td></tr> <tr> <th scope="row">e </th> <td><a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> </td> <td>ln x <a href="#cite_note-adaa-21">[nb 3]</a> </td> <td>log x, loge x </td></tr> <tr> <th scope="row">10 </th> <td><a href="/wiki/Common_logarithm" title="Common logarithm">common logarithm</a> </td> <td>lg x </td> <td>log x, log10 x </td></tr> <tr> <th scope="row">b </th> <td>logarithm to base b </td> <td>logb x </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=8" title="Edit section: History">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_logarithms" title="History of logarithms">History of logarithms</a></div> The history of logarithms in seventeenth-century Europe saw the discovery of a new <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by <a href="/wiki/John_Napier" title="John Napier">John Napier</a> in 1614, in a book titled <a href="/wiki/Mirifici_Logarithmorum_Canonis_Descriptio" title="Mirifici Logarithmorum Canonis Descriptio">Mirifici Logarithmorum Canonis Descriptio</a> (Description of the Wonderful Canon of Logarithms).<a href="#cite_note-22">[19]</a><a href="#cite_note-23">[20]</a> Prior to Napier's invention, there had been other techniques of similar scopes, such as the <a href="/wiki/Prosthaphaeresis" title="Prosthaphaeresis">prosthaphaeresis</a> or the use of tables of progressions, extensively developed by <a href="/wiki/Jost_B%C3%BCrgi" title="Jost Bürgi">Jost Bürgi</a> around 1600.<a href="#cite_note-folkerts-24">[21]</a><a href="#cite_note-25">[22]</a> Napier coined the term for logarithm in Middle Latin, logarithmus, literally meaning 'ratio-number', derived from the Greek logos 'proportion, ratio, word' + arithmos 'number'. The <a href="/wiki/Common_logarithm" title="Common logarithm">common logarithm</a> of a number is the index of that power of ten which equals the number.<a href="#cite_note-26">[23]</a> Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> as the "order of a number".<a href="#cite_note-27">[24]</a> The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation. Some of these methods used tables derived from trigonometric identities.<a href="#cite_note-28">[25]</a> Such methods are called <a href="/wiki/Prosthaphaeresis" title="Prosthaphaeresis">prosthaphaeresis</a>. Invention of the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> now known as the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> began as an attempt to perform a <a href="/wiki/Quadrature_(mathematics)" class="mw-redirect" title="Quadrature (mathematics)">quadrature</a> of a rectangular <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> by <a href="/wiki/Gr%C3%A9goire_de_Saint-Vincent" title="Grégoire de Saint-Vincent">Grégoire de Saint-Vincent</a>, a Belgian Jesuit residing in Prague. Archimedes had written <a href="/wiki/The_Quadrature_of_the_Parabola" class="mw-redirect" title="The Quadrature of the Parabola">The Quadrature of the Parabola</a> in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a <a href="/wiki/Geometric_progression" title="Geometric progression">geometric progression</a> in its <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a> and an <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a> of values, prompted <a href="/wiki/A._A._de_Sarasa" class="mw-redirect" title="A. A. de Sarasa">A. A. de Sarasa</a> to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in <a href="/wiki/Prosthaphaeresis" title="Prosthaphaeresis">prosthaphaeresis</a>, leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Christiaan Huygens</a>, and <a href="/wiki/James_Gregory_(mathematician)" title="James Gregory (mathematician)">James Gregory</a>. The notation Log y was adopted by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> in 1675,<a href="#cite_note-29">[26]</a> and the next year he connected it to the <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int {\frac {dy}{y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mi>y</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int {\frac {dy}{y}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/520963f5a439ba8538bb12cfe16fd0345e641385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.965ex; height:4.176ex;" alt="{\textstyle \int {\frac {dy}{y}}.}"> Before Euler developed his modern conception of complex natural logarithms, <a href="/wiki/Roger_Cotes#Mathematics" title="Roger Cotes">Roger Cotes</a> had a nearly equivalent result when he showed in 1714 that<a href="#cite_note-30">[27]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(\cos \theta +i\sin \theta )=i\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>i</mi> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(\cos \theta +i\sin \theta )=i\theta .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea630241d3f7d3497fb2fcae4dacb2434cc2010" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.371ex; height:2.843ex;" alt="{\displaystyle \log(\cos \theta +i\sin \theta )=i\theta .}"> <div class="mw-heading mw-heading2"><h2 id="Logarithm_tables,_slide_rules,_and_historical_applications">Logarithm tables, slide rules, and historical applications</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=9" title="Edit section: Logarithm tables, slide rules, and historical applications">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Logarithms_Britannica_1797.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Logarithms_Britannica_1797.png/360px-Logarithms_Britannica_1797.png" decoding="async" width="360" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Logarithms_Britannica_1797.png/540px-Logarithms_Britannica_1797.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Logarithms_Britannica_1797.png/720px-Logarithms_Britannica_1797.png 2x" data-file-width="997" data-file-height="354" /></a><figcaption>The 1797 <a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a> explanation of logarithms</figcaption></figure> By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>. They were critical to advances in <a href="/wiki/Surveying" title="Surveying">surveying</a>, <a href="/wiki/Celestial_navigation" title="Celestial navigation">celestial navigation</a>, and other domains. <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> called logarithms <dl><dd><dl><dd>"...[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."<a href="#cite_note-31">[28]</a></dd></dl></dd></dl> As the function f(x) = bx is the inverse function of logb x, it has been called an antilogarithm.<a href="#cite_note-32">[29]</a> Nowadays, this function is more commonly called an <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>. <div class="mw-heading mw-heading3"><h3 id="Log_tables">Log tables</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=10" title="Edit section: Log tables">edit</a>]</div> A key tool that enabled the practical use of logarithms was the <a href="/wiki/Log_table" class="mw-redirect" title="Log table">table of logarithms</a>.<a href="#cite_note-33">[30]</a> The first such table was compiled by <a href="/wiki/Henry_Briggs_(mathematician)" title="Henry Briggs (mathematician)">Henry Briggs</a> in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the <a href="/wiki/Common_logarithm" title="Common logarithm">common logarithms</a> of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of log10 x for any number x in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of x can be separated into an <a href="/wiki/Integer_part" class="mw-redirect" title="Integer part">integer part</a> and a <a href="/wiki/Fractional_part" title="Fractional part">fractional part</a>, known as the characteristic and <a href="/wiki/Mantissa_(logarithm)" class="mw-redirect" title="Mantissa (logarithm)">mantissa</a>. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.<a href="#cite_note-34">[31]</a> The characteristic of 10 · x is one plus the characteristic of x, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\log _{10}3542&=\log _{10}(1000\cdot 3.542)\\&=3+\log _{10}3.542\\&\approx 3+\log _{10}3.54\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>3542</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1000</mn> <mo>⋅</mo> <mn>3.542</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>3.542</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mn>3</mn> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>3.54</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log _{10}3542&=\log _{10}(1000\cdot 3.542)\\&=3+\log _{10}3.542\\&\approx 3+\log _{10}3.54\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d53c38f1bbb1969092aeb2a950a315212ee7f8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:32.018ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\log _{10}3542&=\log _{10}(1000\cdot 3.542)\\&=3+\log _{10}3.542\\&\approx 3+\log _{10}3.54\end{aligned}}}"> Greater accuracy can be obtained by <a href="/wiki/Interpolation" title="Interpolation">interpolation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}3542\approx {}3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>3542</mn> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mn>3</mn> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>3.54</mn> <mo>+</mo> <mn>0.2</mn> <mo stretchy="false">(</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>3.55</mn> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>3.54</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}3542\approx {}3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2394d5653802e38bea193035e2ec105a2399de94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.556ex; height:2.843ex;" alt="{\displaystyle \log _{10}3542\approx {}3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)}"> The value of 10x can be determined by reverse look up in the same table, since the logarithm is a <a href="/wiki/Monotonic_function" title="Monotonic function">monotonic function</a>. <div class="mw-heading mw-heading3"><h3 id="Computations">Computations</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=11" title="Edit section: Computations">edit</a>]</div> The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. The product cd or quotient c/d came from looking up the antilogarithm of the sum or difference, via the same table: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>d</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>c</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>d</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961312f47e633d6119627a74dbbd40fd6b284b9a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:38.785ex; height:2.676ex;" alt="{\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mi>c</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>d</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c9e13cc3cd58305405feeae0eb07d73d26cde7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.997ex; height:4.843ex;" alt="{\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.}"> For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as <a href="/wiki/Prosthaphaeresis" title="Prosthaphaeresis">prosthaphaeresis</a>, which relies on <a href="/wiki/Trigonometric_identities" class="mw-redirect" title="Trigonometric identities">trigonometric identities</a>. Calculations of powers and <a href="/wiki/Nth_root" title="Nth root">roots</a> are reduced to multiplications or divisions and lookups by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>c</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a448af3e9527c481ac770c0d90dba4157a8aa99" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.052ex; height:3.843ex;" alt="{\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </mroot> </mrow> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>d</mi> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>c</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cee4c3ee52a4b250c30c38836d4d58a006ce74c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.737ex; height:4.176ex;" alt="{\displaystyle {\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.}"> Trigonometric calculations were facilitated by tables that contained the common logarithms of <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a>. <div class="mw-heading mw-heading3"><h3 id="Slide_rules">Slide rules</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=12" title="Edit section: Slide rules">edit</a>]</div> Another critical application was the <a href="/wiki/Slide_rule" title="Slide rule">slide rule</a>, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, <a href="/wiki/Gunter%27s_rule" class="mw-redirect" title="Gunter's rule">Gunter's rule</a>, was invented shortly after Napier's invention. <a href="/wiki/William_Oughtred" title="William Oughtred">William Oughtred</a> enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here: <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Slide_rule_example2_with_labels.svg" class="mw-file-description"><img alt="A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Slide_rule_example2_with_labels.svg/550px-Slide_rule_example2_with_labels.svg.png" decoding="async" width="550" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Slide_rule_example2_with_labels.svg/825px-Slide_rule_example2_with_labels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Slide_rule_example2_with_labels.svg/1100px-Slide_rule_example2_with_labels.svg.png 2x" data-file-width="512" data-file-height="119" /></a><figcaption>Schematic depiction of a slide rule. Starting from 2 on the lower scale, add the distance to 3 on the upper scale to reach the product 6. The slide rule works because it is marked such that the distance from 1 to x is proportional to the logarithm of x.</figcaption></figure> For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.<a href="#cite_note-ReferenceA-35">[32]</a> <div class="mw-heading mw-heading2"><h2 id="Analytic_properties">Analytic properties</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=13" title="Edit section: Analytic properties">edit</a>]</div> A deeper study of logarithms requires the concept of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>. A function is a rule that, given one number, produces another number.<a href="#cite_note-36">[33]</a> An example is the function producing the x-th power of b from any real number x, where the base b is a fixed number. This function is written as f(x) = b x. When b is positive and unequal to 1, we show below that f is invertible when considered as a function from the reals to the positive reals. <div class="mw-heading mw-heading3"><h3 id="Existence">Existence</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=14" title="Edit section: Existence">edit</a>]</div> Let b be a positive real number not equal to 1 and let f(x) = b x. It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from the <a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a>.<a href="#cite_note-LangIII.3-37">[34]</a> Now, f is <a href="/wiki/Monotonic_function" title="Monotonic function">strictly increasing</a> (for b > 1), or strictly decreasing (for 0 < b < 1),<a href="#cite_note-LangIV.2-38">[35]</a> is continuous, has domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">, and has range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{>0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>></mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{>0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731b0a191e1eb70161af731d0d567b236457074f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.011ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{>0}}">. Therefore, f is a bijection from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{>0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>></mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{>0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/731b0a191e1eb70161af731d0d567b236457074f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.011ex; height:2.509ex;" alt="{\displaystyle \mathbb {R} _{>0}}">. In other words, for each positive real number y, there is exactly one real number x such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{x}=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{x}=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711753605e98f4d42b75fe61254c3b8f311a5fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.424ex; height:2.676ex;" alt="{\displaystyle b^{x}=y}">. We let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}\colon \mathbb {R} _{>0}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>:</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>></mo> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}\colon \mathbb {R} _{>0}\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6e100b69edcf61555f7a41633c3c34c1a969a97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.246ex; height:2.676ex;" alt="{\displaystyle \log _{b}\colon \mathbb {R} _{>0}\to \mathbb {R} }"> denote the inverse of f. That is, logb y is the unique real number x such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{x}=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{x}=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711753605e98f4d42b75fe61254c3b8f311a5fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.424ex; height:2.676ex;" alt="{\displaystyle b^{x}=y}">. This function is called the base-b logarithm function or logarithmic function (or just logarithm). <div class="mw-heading mw-heading3"><h3 id="Characterization_by_the_product_formula">Characterization by the product formula</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=15" title="Edit section: Characterization by the product formula">edit</a>]</div> The function logb x can also be essentially characterized by the product formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1de92e0c333c4644f8866e417ea56e715324ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.868ex; height:2.843ex;" alt="{\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.}"> More precisely, the logarithm to any base b > 1 is the only <a href="/wiki/Increasing_function" class="mw-redirect" title="Increasing function">increasing function</a> f from the positive reals to the reals satisfying f(b) = 1 and<a href="#cite_note-39">[36]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(xy)=f(x)+f(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(xy)=f(x)+f(y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee6f0eb6e355d16f673a0d4a21705e24a227008" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.82ex; height:2.843ex;" alt="{\displaystyle f(xy)=f(x)+f(y).}"> <div class="mw-heading mw-heading3"><h3 id="Graph_of_the_logarithm_function">Graph of the logarithm function</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=16" title="Edit section: Graph of the logarithm function">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Logarithm_inversefunctiontoexp.svg" class="mw-file-description"><img alt="The graphs of two functions." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Logarithm_inversefunctiontoexp.svg/220px-Logarithm_inversefunctiontoexp.svg.png" decoding="async" width="220" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Logarithm_inversefunctiontoexp.svg/330px-Logarithm_inversefunctiontoexp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Logarithm_inversefunctiontoexp.svg/440px-Logarithm_inversefunctiontoexp.svg.png 2x" data-file-width="240" data-file-height="279" /></a><figcaption>The graph of the logarithm function logb (x) (blue) is obtained by <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflecting</a> the graph of the function bx (red) at the diagonal line (x = y).</figcaption></figure> As discussed above, the function logb is the inverse to the exponential function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto b^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto b^{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/873987f9618fe2c30ce4e72cbd1a967ff759c1d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.114ex; height:2.343ex;" alt="{\displaystyle x\mapsto b^{x}}">. Therefore, their <a href="/wiki/Graph_of_a_function" title="Graph of a function">graphs</a> correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line x = y), as shown at the right: a point (t, u = bt) on the graph of f yields a point (u, t = logb u) on the graph of the logarithm and vice versa. As a consequence, logb (x) <a href="/wiki/Divergent_sequence" class="mw-redirect" title="Divergent sequence">diverges to infinity</a> (gets bigger than any given number) if x grows to infinity, provided that b is greater than one. In that case, logb(x) is an <a href="/wiki/Increasing_function" class="mw-redirect" title="Increasing function">increasing function</a>. For b < 1, logb (x) tends to minus infinity instead. When x approaches zero, logb x goes to minus infinity for b > 1 (plus infinity for b < 1, respectively). <div class="mw-heading mw-heading3"><h3 id="Derivative_and_antiderivative">Derivative and antiderivative</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=17" title="Edit section: Derivative and antiderivative">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Logarithm_derivative.svg" class="mw-file-description"><img alt="A graph of the logarithm function and a line touching it in one point." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Logarithm_derivative.svg/220px-Logarithm_derivative.svg.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Logarithm_derivative.svg/330px-Logarithm_derivative.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Logarithm_derivative.svg/440px-Logarithm_derivative.svg.png 2x" data-file-width="375" data-file-height="243" /></a><figcaption>The graph of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> (green) and its tangent at x = 1.5 (black)</figcaption></figure> Analytic properties of functions pass to their inverses.<a href="#cite_note-LangIII.3-37">[34]</a> Thus, as f(x) = bx is a continuous and <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a>, so is logb y. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the <a href="/wiki/Derivative" title="Derivative">derivative</a> of f(x) evaluates to ln(b) bx by the properties of the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> implies that the derivative of logb x is given by<a href="#cite_note-LangIV.2-38">[35]</a><a href="#cite_note-40">[37]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c36ba5e891172556f0477a117831310bcd38869c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.017ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.}"> That is, the <a href="/wiki/Slope" title="Slope">slope</a> of the <a href="/wiki/Tangent" title="Tangent">tangent</a> touching the graph of the base-b logarithm at the point (x, logb (x)) equals 1/(x ln(b)). The derivative of ln(x) is 1/x; this implies that ln(x) is the unique <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> of 1/x that has the value 0 for x = 1. It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">constant e</a>. The derivative with a generalized functional argument f(x) is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {f'(x)}{f(x)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {f'(x)}{f(x)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06fb9da0538b2a1eefa892ebbfbc3fffdc98bc0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.238ex; height:6.509ex;" alt="{\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {f'(x)}{f(x)}}.}"> The quotient at the right hand side is called the <a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">logarithmic derivative</a> of f. Computing f'(x) by means of the derivative of ln(f(x)) is known as <a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">logarithmic differentiation</a>.<a href="#cite_note-41">[38]</a> The antiderivative of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> ln(x) is:<a href="#cite_note-42">[39]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcaf5c8b14b232de9ff79e9ae0960ea4966bd10" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.909ex; height:5.676ex;" alt="{\displaystyle \int \ln(x)\,dx=x\ln(x)-x+C.}"> <a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">Related formulas</a>, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.<a href="#cite_note-43">[40]</a> <div class="mw-heading mw-heading3"><h3 id="Integral_representation_of_the_natural_logarithm">Integral representation of the natural logarithm</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=18" title="Edit section: Integral representation of the natural logarithm">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Natural_logarithm_integral.svg" class="mw-file-description"><img alt="A hyperbola with part of the area underneath shaded in grey." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Natural_logarithm_integral.svg/220px-Natural_logarithm_integral.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Natural_logarithm_integral.svg/330px-Natural_logarithm_integral.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Natural_logarithm_integral.svg/440px-Natural_logarithm_integral.svg.png 2x" data-file-width="601" data-file-height="301" /></a><figcaption>The <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> of t is the shaded area underneath the graph of the function f(x) = 1/x.</figcaption></figure> The <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> of t can be defined as the <a href="/wiki/Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln t=\int _{1}^{t}{\frac {1}{x}}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln t=\int _{1}^{t}{\frac {1}{x}}\,dx.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab90a884aa3cc91d3cdcfb9b39992598131621e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.687ex; height:6.176ex;" alt="{\displaystyle \ln t=\int _{1}^{t}{\frac {1}{x}}\,dx.}"> This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, ln(t) equals the area between the x-axis and the graph of the function 1/x, ranging from x = 1 to x = t. This is a consequence of the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> and the fact that the derivative of ln(x) is 1/x. Product and power logarithm formulas can be derived from this definition.<a href="#cite_note-44">[41]</a> For example, the product formula ln(tu) = ln(t) + ln(u) is deduced as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln(tu)&=\int _{1}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(1)}{=}}\int _{1}^{t}{\frac {1}{x}}\,dx+\int _{t}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(2)}{=}}\ln(t)+\int _{1}^{u}{\frac {1}{w}}\,dw\\&=\ln(t)+\ln(u).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>u</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mover> </mrow> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>u</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mover> </mrow> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>w</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>w</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln(tu)&=\int _{1}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(1)}{=}}\int _{1}^{t}{\frac {1}{x}}\,dx+\int _{t}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(2)}{=}}\ln(t)+\int _{1}^{u}{\frac {1}{w}}\,dw\\&=\ln(t)+\ln(u).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed3e6eef0b8687b84a9776500c48a2793b25017" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:30.49ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}\ln(tu)&=\int _{1}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(1)}{=}}\int _{1}^{t}{\frac {1}{x}}\,dx+\int _{t}^{tu}{\frac {1}{x}}\,dx\\&{\stackrel {(2)}{=}}\ln(t)+\int _{1}^{u}{\frac {1}{w}}\,dw\\&=\ln(t)+\ln(u).\end{aligned}}}"> The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (w = x/t). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor t and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function f(x) = 1/x again. Therefore, the left hand blue area, which is the integral of f(x) from t to tu is the same as the integral from 1 to u. This justifies the equality (2) with a more geometric proof. <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Natural_logarithm_product_formula_proven_geometrically.svg" class="mw-file-description"><img alt="The hyperbola depicted twice. The area underneath is split into different parts." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Natural_logarithm_product_formula_proven_geometrically.svg/500px-Natural_logarithm_product_formula_proven_geometrically.svg.png" decoding="async" width="500" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Natural_logarithm_product_formula_proven_geometrically.svg/750px-Natural_logarithm_product_formula_proven_geometrically.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Natural_logarithm_product_formula_proven_geometrically.svg/1000px-Natural_logarithm_product_formula_proven_geometrically.svg.png 2x" data-file-width="1353" data-file-height="304" /></a><figcaption>A visual proof of the product formula of the natural logarithm</figcaption></figure> The power formula ln(tr) = r ln(t) may be derived in a similar way: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln(t^{r})&=\int _{1}^{t^{r}}{\frac {1}{x}}dx\\&=\int _{1}^{t}{\frac {1}{w^{r}}}\left(rw^{r-1}\,dw\right)\\&=r\int _{1}^{t}{\frac {1}{w}}\,dw\\&=r\ln(t).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>w</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>w</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>w</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln(t^{r})&=\int _{1}^{t^{r}}{\frac {1}{x}}dx\\&=\int _{1}^{t}{\frac {1}{w^{r}}}\left(rw^{r-1}\,dw\right)\\&=r\int _{1}^{t}{\frac {1}{w}}\,dw\\&=r\ln(t).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863f225bdfee5e6256df66cc7b5c44ba2fde8e2e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:28.134ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}\ln(t^{r})&=\int _{1}^{t^{r}}{\frac {1}{x}}dx\\&=\int _{1}^{t}{\frac {1}{w^{r}}}\left(rw^{r-1}\,dw\right)\\&=r\int _{1}^{t}{\frac {1}{w}}\,dw\\&=r\ln(t).\end{aligned}}}"> The second equality uses a change of variables (<a href="/wiki/Integration_by_substitution" title="Integration by substitution">integration by substitution</a>), w = x1/r. The sum over the reciprocals of natural numbers, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f50a7390e77e5beed851612314d2d03991d564" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.01ex; height:6.843ex;" alt="{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}},}"> is called the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a>. It is closely tied to the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>: as n tends to <a href="/wiki/Infinity" title="Infinity">infinity</a>, the difference, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}-\ln(n),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}-\ln(n),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f1edf2104b89524c509d6cb9ea1a667251d3ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.42ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}{\frac {1}{k}}-\ln(n),}"> <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">converges</a> (i.e. gets arbitrarily close) to a number known as the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a> γ = 0.5772.... This relation aids in analyzing the performance of algorithms such as <a href="/wiki/Quicksort" title="Quicksort">quicksort</a>.<a href="#cite_note-45">[42]</a> <div class="mw-heading mw-heading3"><h3 id="Transcendence_of_the_logarithm">Transcendence of the logarithm</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=19" title="Edit section: Transcendence of the logarithm">edit</a>]</div> <a href="/wiki/Real_number" title="Real number">Real numbers</a> that are not <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic</a> are called <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>;<a href="#cite_note-46">[43]</a> for example, <a href="/wiki/Pi" title="Pi">π</a> and <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a> are such numbers, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2-{\sqrt {3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2-{\sqrt {3}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b2a724c326f7f59f71baa9788cf455a0609bdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.425ex; height:4.843ex;" alt="{\displaystyle {\sqrt {2-{\sqrt {3}}}}}"> is not. <a href="/wiki/Almost_all" title="Almost all">Almost all</a> real numbers are transcendental. The logarithm is an example of a <a href="/wiki/Transcendental_function" title="Transcendental function">transcendental function</a>. The <a href="/wiki/Gelfond%E2%80%93Schneider_theorem" title="Gelfond–Schneider theorem">Gelfond–Schneider theorem</a> asserts that logarithms usually take transcendental, i.e. "difficult" values.<a href="#cite_note-47">[44]</a> <div class="mw-heading mw-heading2"><h2 id="Calculation">Calculation</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=20" title="Edit section: Calculation">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Logarithm_keys.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Logarithm_keys.jpg/220px-Logarithm_keys.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Logarithm_keys.jpg/330px-Logarithm_keys.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Logarithm_keys.jpg/440px-Logarithm_keys.jpg 2x" data-file-width="1882" data-file-height="1411" /></a><figcaption>The logarithm keys (LOG for base 10 and LN for base e) on a <a href="/wiki/TI-83_series" title="TI-83 series">TI-83 Plus</a> graphing calculator</figcaption></figure> Logarithms are easy to compute in some cases, such as log10 (1000) = 3. In general, logarithms can be calculated using <a href="/wiki/Power_series" title="Power series">power series</a> or the <a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">arithmetic–geometric mean</a>, or be retrieved from a precalculated <a href="/wiki/Logarithm_table" class="mw-redirect" title="Logarithm table">logarithm table</a> that provides a fixed precision.<a href="#cite_note-48">[45]</a><a href="#cite_note-49">[46]</a> <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a>, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently.<a href="#cite_note-50">[47]</a> Using look-up tables, <a href="/wiki/CORDIC" title="CORDIC">CORDIC</a>-like methods can be used to compute logarithms by using only the operations of addition and <a href="/wiki/Arithmetic_shift" title="Arithmetic shift">bit shifts</a>.<a href="#cite_note-51">[48]</a><a href="#cite_note-52">[49]</a> Moreover, the <a href="/wiki/Binary_logarithm#Algorithm" title="Binary logarithm">binary logarithm algorithm</a> calculates lb(x) <a href="/wiki/Recursion" title="Recursion">recursively</a>, based on repeated squarings of x, taking advantage of the relation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}\left(x^{2}\right)=2\log _{2}|x|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}\left(x^{2}\right)=2\log _{2}|x|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d064349793d675f2707a5142e3530bd32facbdc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.871ex; height:3.343ex;" alt="{\displaystyle \log _{2}\left(x^{2}\right)=2\log _{2}|x|.}"> <div class="mw-heading mw-heading3"><h3 id="Power_series">Power series</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=21" title="Edit section: Power series">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Taylor_series">Taylor series</h4>[<a href="/w/index.php?title=Logarithm&action=edit&section=22" title="Edit section: Taylor series">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Taylor_approximation_of_natural_logarithm.gif" class="mw-file-description"><img alt="An animation showing increasingly good approximations of the logarithm graph." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Taylor_approximation_of_natural_logarithm.gif/220px-Taylor_approximation_of_natural_logarithm.gif" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/02/Taylor_approximation_of_natural_logarithm.gif 1.5x" data-file-width="300" data-file-height="185" /></a><figcaption>The Taylor series of ln(z) centered at z = 1. The animation shows the first 10 approximations along with the 99th and 100th. The approximations do not converge beyond a distance of 1 from the center.</figcaption></figure> For any real number z that satisfies 0 < z ≤ 2, the following formula holds:<a href="#cite_note-53">[nb 4]</a><a href="#cite_note-AbramowitzStegunp.68-54">[50]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln(z)&={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {(z-1)^{k}}{k}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln(z)&={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {(z-1)^{k}}{k}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c57a960210bc2a905785a5be6ba3d4ef2cfa62" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.765ex; margin-bottom: -0.24ex; width:57.934ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\ln(z)&={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {(z-1)^{k}}{k}}.\end{aligned}}}"> Equating the function ln(z) to this infinite sum (<a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a>) is shorthand for saying that the function can be approximated to a more and more accurate value by the following expressions (known as <a href="/wiki/Partial_sum" class="mw-redirect" title="Partial sum">partial sums</a>): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (z-1),\ \ (z-1)-{\frac {(z-1)^{2}}{2}},\ \ (z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}},\ \ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (z-1),\ \ (z-1)-{\frac {(z-1)^{2}}{2}},\ \ (z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}},\ \ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6741aff58f21f04c60cd1f9e7cc2ed895d990327" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:64.709ex; height:5.843ex;" alt="{\displaystyle (z-1),\ \ (z-1)-{\frac {(z-1)^{2}}{2}},\ \ (z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}},\ \ldots }"> For example, with z = 1.5 the third approximation yields 0.4167, which is about 0.011 greater than ln(1.5) = 0.405465, and the ninth approximation yields 0.40553, which is only about 0.0001 greater. The nth partial sum can approximate ln(z) with arbitrary precision, provided the number of summands n is large enough. In elementary calculus, the series is said to <a href="/wiki/Convergent_series" title="Convergent series">converge</a> to the function ln(z), and the function is the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of the series. It is the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> at z = 1. The Taylor series of ln(z) provides a particularly useful approximation to ln(1 + z) when z is small, |z| < 1, since then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(1+z)=z-{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}-\cdots \approx z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>≈</mo> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(1+z)=z-{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}-\cdots \approx z.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6eb8f63f6959cb0e21a2ea79b0edd5d698f1528" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.066ex; height:5.676ex;" alt="{\displaystyle \ln(1+z)=z-{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}-\cdots \approx z.}"> For example, with z = 0.1 the first-order approximation gives ln(1.1) ≈ 0.1, which is less than 5% off the correct value 0.0953. <div class="mw-heading mw-heading4"><h4 id="Inverse_hyperbolic_tangent">Inverse hyperbolic tangent</h4>[<a href="/w/index.php?title=Logarithm&action=edit&section=23" title="Edit section: Inverse hyperbolic tangent">edit</a>]</div> Another series is based on the <a href="/wiki/Area_hyperbolic_tangent#Inverse_hyperbolic_tangent" class="mw-redirect" title="Area hyperbolic tangent">inverse hyperbolic tangent</a> function: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(z)=2\cdot \operatorname {artanh} \,{\frac {z-1}{z+1}}=2\left({\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mi>artanh</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(z)=2\cdot \operatorname {artanh} \,{\frac {z-1}{z+1}}=2\left({\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4774ed055db87556b991ffc8dcf5bd795f823c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:75.541ex; height:7.509ex;" alt="{\displaystyle \ln(z)=2\cdot \operatorname {artanh} \,{\frac {z-1}{z+1}}=2\left({\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right),}"> for any real number z > 0.<a href="#cite_note-55">[nb 5]</a><a href="#cite_note-AbramowitzStegunp.68-54">[50]</a> Using <a href="/wiki/Sigma_notation" class="mw-redirect" title="Sigma notation">sigma notation</a>, this is also written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(z)=2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {z-1}{z+1}}\right)^{2k+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(z)=2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {z-1}{z+1}}\right)^{2k+1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d9729501b26eb85764942cb112cc9885b1a6cca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.446ex; height:7.343ex;" alt="{\displaystyle \ln(z)=2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {z-1}{z+1}}\right)^{2k+1}.}"> This series can be derived from the above Taylor series. It converges quicker than the Taylor series, especially if z is close to 1. For example, for z = 1.5, the first three terms of the second series approximate ln(1.5) with an error of about 3×10−6. The quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation y ≈ ln(z) and putting <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {z}{\exp(y)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {z}{\exp(y)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70dc2d5fb51bc065e7662ba91fe25996896dfa2e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.842ex; height:5.509ex;" alt="{\displaystyle A={\frac {z}{\exp(y)}},}"> the logarithm of z is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(z)=y+\ln(A).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(z)=y+\ln(A).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea935537fad8ed9632a4e0eaa453c172906606c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.07ex; height:2.843ex;" alt="{\displaystyle \ln(z)=y+\ln(A).}"> The better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently. A can be calculated using the <a href="/wiki/Exponential_function" title="Exponential function">exponential series</a>, which converges quickly provided y is not too large. Calculating the logarithm of larger z can be reduced to smaller values of z by writing z = a · 10b, so that ln(z) = ln(a) + b · ln(10). A closely related method can be used to compute the logarithm of integers. Putting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle z={\frac {n+1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle z={\frac {n+1}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4197236e535dc6467df2570a52e72ec095f7fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.109ex; height:3.509ex;" alt="{\displaystyle \textstyle z={\frac {n+1}{n}}}"> in the above series, it follows that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(n+1)=\ln(n)+2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {1}{2n+1}}\right)^{2k+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(n+1)=\ln(n)+2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {1}{2n+1}}\right)^{2k+1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306cedcb8ef32fe57d535b3c27b2ae6af9b13326" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.208ex; height:7.343ex;" alt="{\displaystyle \ln(n+1)=\ln(n)+2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {1}{2n+1}}\right)^{2k+1}.}"> If the logarithm of a large integer n is known, then this series yields a fast converging series for log(n+1), with a <a href="/wiki/Rate_of_convergence" title="Rate of convergence">rate of convergence</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left({\frac {1}{2n+1}}\right)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left({\frac {1}{2n+1}}\right)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c094c1c61f349b1216928c33fe29aa61860626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.575ex; height:5.176ex;" alt="{\textstyle \left({\frac {1}{2n+1}}\right)^{2}}">. <div class="mw-heading mw-heading3"><h3 id="Arithmetic–geometric_mean_approximation">Arithmetic–geometric mean approximation</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=24" title="Edit section: Arithmetic–geometric mean approximation">edit</a>]</div> The <a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">arithmetic–geometric mean</a> yields high-precision approximations of the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work ln(x) is approximated to a precision of 2−p (or p precise bits) by the following formula (due to <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>):<a href="#cite_note-56">[51]</a><a href="#cite_note-57">[52]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)\approx {\frac {\pi }{2\,\mathrm {M} \!\left(1,2^{2-m}/x\right)}}-m\ln(2).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>m</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)\approx {\frac {\pi }{2\,\mathrm {M} \!\left(1,2^{2-m}/x\right)}}-m\ln(2).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec803ea11552f9cdfd17caf1b39cf8e7a8e84184" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.276ex; height:6.009ex;" alt="{\displaystyle \ln(x)\approx {\frac {\pi }{2\,\mathrm {M} \!\left(1,2^{2-m}/x\right)}}-m\ln(2).}"> Here M(x, y) denotes the <a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">arithmetic–geometric mean</a> of x and y. It is obtained by repeatedly calculating the average (x + y)/2 (<a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {xy}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mi>y</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {xy}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966df76a5bc207e11606ad5c8ea6788d6a838c47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.421ex; height:3.009ex;" alt="{\textstyle {\sqrt {xy}}}"> (<a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a>) of x and y then let those two numbers become the next x and y. The two numbers quickly converge to a common limit which is the value of M(x, y). m is chosen such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,2^{m}>2^{p/2}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thinmathspace" /> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>></mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\,2^{m}>2^{p/2}.\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80443424cc40c061dba53d32f86d2d8169aa1983" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.552ex; height:2.843ex;" alt="{\displaystyle x\,2^{m}>2^{p/2}.\,}"> to ensure the required precision. A larger m makes the M(x, y) calculation take more steps (the initial x and y are farther apart so it takes more steps to converge) but gives more precision. The constants π and ln(2) can be calculated with quickly converging series. <div class="mw-heading mw-heading3"><h3 id="Feynman's_algorithm">Feynman's algorithm</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=25" title="Edit section: Feynman's algorithm">edit</a>]</div> While at <a href="/wiki/Los_Alamos_National_Laboratory" title="Los Alamos National Laboratory">Los Alamos National Laboratory</a> working on the <a href="/wiki/Manhattan_Project" title="Manhattan Project">Manhattan Project</a>, <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> developed a bit-processing algorithm to compute the logarithm that is similar to long division and was later used in the <a href="/wiki/Connection_Machine" title="Connection Machine">Connection Machine</a>. The algorithm relies on the fact that every real number x where 1 < x < 2 can be represented as a product of distinct factors of the form 1 + 2−k. The algorithm sequentially builds that product P, starting with P = 1 and k = 1: if P · (1 + 2−k) < x, then it changes P to P · (1 + 2−k). It then increases <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> by one regardless. The algorithm stops when k is large enough to give the desired accuracy. Because log(x) is the sum of the terms of the form log(1 + 2−k) corresponding to those k for which the factor 1 + 2−k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2−k) for all k. Any base may be used for the logarithm table.<a href="#cite_note-58">[53]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=26" title="Edit section: Applications">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NautilusCutawayLogarithmicSpiral.jpg" class="mw-file-description"><img alt="A photograph of a nautilus' shell." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/220px-NautilusCutawayLogarithmicSpiral.jpg" decoding="async" width="220" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/330px-NautilusCutawayLogarithmicSpiral.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/NautilusCutawayLogarithmicSpiral.jpg/440px-NautilusCutawayLogarithmicSpiral.jpg 2x" data-file-width="2240" data-file-height="1693" /></a><figcaption>A <a href="/wiki/Nautilus" title="Nautilus">nautilus</a> shell displaying a logarithmic spiral</figcaption></figure> Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of <a href="/wiki/Scale_invariance" title="Scale invariance">scale invariance</a>. For example, each chamber of the shell of a <a href="/wiki/Nautilus" title="Nautilus">nautilus</a> is an approximate copy of the next one, scaled by a constant factor. This gives rise to a <a href="/wiki/Logarithmic_spiral" title="Logarithmic spiral">logarithmic spiral</a>.<a href="#cite_note-59">[54]</a> <a href="/wiki/Benford%27s_law" title="Benford's law">Benford's law</a> on the distribution of leading digits can also be explained by scale invariance.<a href="#cite_note-60">[55]</a> Logarithms are also linked to <a href="/wiki/Self-similarity" title="Self-similarity">self-similarity</a>. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.<a href="#cite_note-61">[56]</a> The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms. <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">Logarithmic scales</a> are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function log(x) grows very slowly for large x, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the <a href="/wiki/Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Tsiolkovsky rocket equation</a>, the <a href="/wiki/Fenske_equation" title="Fenske equation">Fenske equation</a>, or the <a href="/wiki/Nernst_equation" title="Nernst equation">Nernst equation</a>. <div class="mw-heading mw-heading3"><h3 id="Logarithmic_scale">Logarithmic scale</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=27" title="Edit section: Logarithmic scale">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Logarithmic_scale" title="Logarithmic scale">Logarithmic scale</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Germany_Hyperinflation.svg" class="mw-file-description"><img alt="A graph of the value of one mark over time. The line showing its value is increasing very quickly, even with logarithmic scale." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Germany_Hyperinflation.svg/220px-Germany_Hyperinflation.svg.png" decoding="async" width="220" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Germany_Hyperinflation.svg/330px-Germany_Hyperinflation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Germany_Hyperinflation.svg/440px-Germany_Hyperinflation.svg.png 2x" data-file-width="509" data-file-height="594" /></a><figcaption>A logarithmic chart depicting the value of one <a href="/wiki/German_gold_mark" class="mw-redirect" title="German gold mark">Goldmark</a> in <a href="/wiki/German_Papiermark" class="mw-redirect" title="German Papiermark">Papiermarks</a> during the <a href="/wiki/Inflation_in_the_Weimar_Republic" class="mw-redirect" title="Inflation in the Weimar Republic">German hyperinflation in the 1920s</a></figcaption></figure> Scientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the <a href="/wiki/Decibel" title="Decibel">decibel</a> is a <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit of measurement</a> associated with <a href="/wiki/Logarithmic-scale" class="mw-redirect" title="Logarithmic-scale">logarithmic-scale</a> <a href="/wiki/Level_quantity" class="mw-redirect" title="Level quantity">quantities</a>. It is based on the common logarithm of <a href="/wiki/Ratio" title="Ratio">ratios</a>—10 times the common logarithm of a <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> ratio or 20 times the common logarithm of a <a href="/wiki/Voltage" title="Voltage">voltage</a> ratio. It is used to quantify the attenuation or amplification of electrical signals,<a href="#cite_note-62">[57]</a> to describe power levels of sounds in <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>,<a href="#cite_note-63">[58]</a> and the <a href="/wiki/Absorbance" title="Absorbance">absorbance</a> of light in the fields of <a href="/wiki/Spectrometer" title="Spectrometer">spectrometry</a> and <a href="/wiki/Optics" title="Optics">optics</a>. The <a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a> describing the amount of unwanted <a href="/wiki/Noise_(electronic)" class="mw-redirect" title="Noise (electronic)">noise</a> in relation to a (meaningful) <a href="/wiki/Signal_(information_theory)" class="mw-redirect" title="Signal (information theory)">signal</a> is also measured in decibels.<a href="#cite_note-64">[59]</a> In a similar vein, the <a href="/wiki/Peak_signal-to-noise_ratio" title="Peak signal-to-noise ratio">peak signal-to-noise ratio</a> is commonly used to assess the quality of sound and <a href="/wiki/Image_compression" title="Image compression">image compression</a> methods using the logarithm.<a href="#cite_note-65">[60]</a> The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the <a href="/wiki/Moment_magnitude_scale" title="Moment magnitude scale">moment magnitude scale</a> or the <a href="/wiki/Richter_magnitude_scale" class="mw-redirect" title="Richter magnitude scale">Richter magnitude scale</a>. For example, a 5.0 earthquake releases 32 times (101.5) and a 6.0 releases 1000 times (103) the energy of a 4.0.<a href="#cite_note-66">[61]</a> <a href="/wiki/Apparent_magnitude" title="Apparent magnitude">Apparent magnitude</a> measures the brightness of stars logarithmically.<a href="#cite_note-67">[62]</a> In <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> the negative of the decimal logarithm, the decimal <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>cologarithm, is indicated by the letter p.<a href="#cite_note-Jens-68">[63]</a> For instance, <a href="/wiki/PH" title="PH">pH</a> is the decimal cologarithm of the <a href="/wiki/Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a> of <a href="/wiki/Hydronium" title="Hydronium">hydronium</a> ions (the form <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a> <a href="/wiki/Ion" title="Ion">ions</a> H+ take in water).<a href="#cite_note-69">[64]</a> The activity of hydronium ions in neutral water is 10−7 <a href="/wiki/Molar_concentration" title="Molar concentration">mol·L−1</a>, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 104 of the activity, that is, vinegar's hydronium ion activity is about 10−3 mol·L−1. <a href="/wiki/Semi-log_plot" title="Semi-log plot">Semilog</a> (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, <a href="/wiki/Exponential_function" title="Exponential function">exponential functions</a> of the form f(x) = a · bx appear as straight lines with <a href="/wiki/Slope" title="Slope">slope</a> equal to the logarithm of b. <a href="/wiki/Log-log_plot" class="mw-redirect" title="Log-log plot">Log-log</a> graphs scale both axes logarithmically, which causes functions of the form f(x) = a · xk to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing <a href="/wiki/Power_law" title="Power law">power laws</a>.<a href="#cite_note-70">[65]</a> <div class="mw-heading mw-heading3"><h3 id="Psychology">Psychology</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=28" title="Edit section: Psychology">edit</a>]</div> Logarithms occur in several laws describing <a href="/wiki/Human_perception" class="mw-redirect" title="Human perception">human perception</a>:<a href="#cite_note-71">[66]</a><a href="#cite_note-72">[67]</a> <a href="/wiki/Hick%27s_law" title="Hick's law">Hick's law</a> proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have.<a href="#cite_note-73">[68]</a> <a href="/wiki/Fitts%27s_law" title="Fitts's law">Fitts's law</a> predicts that the time required to rapidly move to a target area is a logarithmic function of the ratio between the distance to a target and the size of the target.<a href="#cite_note-74">[69]</a> In <a href="/wiki/Psychophysics" title="Psychophysics">psychophysics</a>, the <a href="/wiki/Weber%E2%80%93Fechner_law" title="Weber–Fechner law">Weber–Fechner law</a> proposes a logarithmic relationship between <a href="/wiki/Stimulus_(psychology)" title="Stimulus (psychology)">stimulus</a> and <a href="/wiki/Sensation_(psychology)" class="mw-redirect" title="Sensation (psychology)">sensation</a> such as the actual vs. the perceived weight of an item a person is carrying.<a href="#cite_note-75">[70]</a> (This "law", however, is less realistic than more recent models, such as <a href="/wiki/Stevens%27s_power_law" title="Stevens's power law">Stevens's power law</a>.<a href="#cite_note-76">[71]</a>) Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.<a href="#cite_note-77">[72]</a><a href="#cite_note-78">[73]</a> <div class="mw-heading mw-heading3"><h3 id="Probability_theory_and_statistics">Probability theory and statistics</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=29" title="Edit section: Probability theory and statistics">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PDF-log_normal_distributions.svg" class="mw-file-description"><img alt="Three asymmetric PDF curves" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/PDF-log_normal_distributions.svg/220px-PDF-log_normal_distributions.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/PDF-log_normal_distributions.svg/330px-PDF-log_normal_distributions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/PDF-log_normal_distributions.svg/440px-PDF-log_normal_distributions.svg.png 2x" data-file-width="390" data-file-height="390" /></a><figcaption>Three <a href="/wiki/Probability_density_function" title="Probability density function">probability density functions</a> (PDF) of random variables with log-normal distributions. The location parameter μ, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Benfords_law_illustrated_by_world%27s_countries_population.svg" class="mw-file-description"><img alt="A bar chart and a superimposed second chart. The two differ slightly, but both decrease in a similar fashion." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Benfords_law_illustrated_by_world%27s_countries_population.svg/220px-Benfords_law_illustrated_by_world%27s_countries_population.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Benfords_law_illustrated_by_world%27s_countries_population.svg/330px-Benfords_law_illustrated_by_world%27s_countries_population.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Benfords_law_illustrated_by_world%27s_countries_population.svg/440px-Benfords_law_illustrated_by_world%27s_countries_population.svg.png 2x" data-file-width="768" data-file-height="576" /></a><figcaption>Distribution of first digits (in %, red bars) in the <a href="/wiki/List_of_countries_by_population" class="mw-redirect" title="List of countries by population">population of the 237 countries</a> of the world. Black dots indicate the distribution predicted by Benford's law.</figcaption></figure> Logarithms arise in <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>: the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> dictates that, for a <a href="/wiki/Fair_coin" title="Fair coin">fair coin</a>, as the number of coin-tosses increases to infinity, the observed proportion of heads <a href="/wiki/Binomial_distribution" title="Binomial distribution">approaches one-half</a>. The fluctuations of this proportion about one-half are described by the <a href="/wiki/Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">law of the iterated logarithm</a>.<a href="#cite_note-79">[74]</a> Logarithms also occur in <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distributions</a>. When the logarithm of a <a href="/wiki/Random_variable" title="Random variable">random variable</a> has a <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a>, the variable is said to have a log-normal distribution.<a href="#cite_note-80">[75]</a> Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.<a href="#cite_note-81">[76]</a> Logarithms are used for <a href="/wiki/Maximum-likelihood_estimation" class="mw-redirect" title="Maximum-likelihood estimation">maximum-likelihood estimation</a> of parametric <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a>. For such a model, the <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a> depends on at least one <a href="/wiki/Parametric_model" title="Parametric model">parameter</a> that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "log likelihood"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for <a href="/wiki/Independence_(probability)" class="mw-redirect" title="Independence (probability)">independent</a> random variables.<a href="#cite_note-82">[77]</a> <a href="/wiki/Benford%27s_law" title="Benford's law">Benford's law</a> describes the occurrence of digits in many <a href="/wiki/Data_set" title="Data set">data sets</a>, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is d (from 1 to 9) equals log10 (d + 1) − log10 (d), regardless of the unit of measurement.<a href="#cite_note-83">[78]</a> Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.<a href="#cite_note-84">[79]</a> The <a href="/wiki/Logarithm_transformation" class="mw-redirect" title="Logarithm transformation">logarithm transformation</a> is a type of <a href="/wiki/Data_transformation_(statistics)" title="Data transformation (statistics)">data transformation</a> used to bring the empirical distribution closer to the assumed one. <div class="mw-heading mw-heading3"><h3 id="Computational_complexity">Computational complexity</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=30" title="Edit section: Computational complexity">edit</a>]</div> <a href="/wiki/Analysis_of_algorithms" title="Analysis of algorithms">Analysis of algorithms</a> is a branch of <a href="/wiki/Computer_science" title="Computer science">computer science</a> that studies the <a href="/wiki/Time_complexity" title="Time complexity">performance</a> of <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> (computer programs solving a certain problem).<a href="#cite_note-Wegener-85">[80]</a> Logarithms are valuable for describing algorithms that <a href="/wiki/Divide_and_conquer_algorithm" class="mw-redirect" title="Divide and conquer algorithm">divide a problem</a> into smaller ones, and join the solutions of the subproblems.<a href="#cite_note-86">[81]</a> For example, to find a number in a sorted list, the <a href="/wiki/Binary_search_algorithm" class="mw-redirect" title="Binary search algorithm">binary search algorithm</a> checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, log2 (N) comparisons, where N is the list's length.<a href="#cite_note-87">[82]</a> Similarly, the <a href="/wiki/Merge_sort" title="Merge sort">merge sort</a> algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time <a href="/wiki/Big_O_notation" title="Big O notation">approximately proportional to</a> N · log(N).<a href="#cite_note-88">[83]</a> The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard <a href="/wiki/Uniform_cost_model" class="mw-redirect" title="Uniform cost model">uniform cost model</a>.<a href="#cite_note-Wegener20-89">[84]</a> A function f(x) is said to <a href="/wiki/Logarithmic_growth" title="Logarithmic growth">grow logarithmically</a> if f(x) is (exactly or approximately) proportional to the logarithm of x. (Biological descriptions of organism growth, however, use this term for an exponential function.<a href="#cite_note-90">[85]</a>) For example, any <a href="/wiki/Natural_number" title="Natural number">natural number</a> N can be represented in <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary form</a> in no more than log2 N + 1 <a href="/wiki/Bit" title="Bit">bits</a>. In other words, the amount of <a href="/wiki/Memory_(computing)" class="mw-redirect" title="Memory (computing)">memory</a> needed to store N grows logarithmically with N. <div class="mw-heading mw-heading3"><h3 id="Entropy_and_chaos">Entropy and chaos</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=31" title="Edit section: Entropy and chaos">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Chaotic_Bunimovich_stadium.svg" class="mw-file-description"><img alt="An oval shape with the trajectories of two particles." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Chaotic_Bunimovich_stadium.svg/220px-Chaotic_Bunimovich_stadium.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Chaotic_Bunimovich_stadium.svg/330px-Chaotic_Bunimovich_stadium.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Chaotic_Bunimovich_stadium.svg/440px-Chaotic_Bunimovich_stadium.svg.png 2x" data-file-width="758" data-file-height="379" /></a><figcaption><a href="/wiki/Dynamical_billiards" title="Dynamical billiards">Billiards</a> on an oval <a href="/wiki/Billiard_table" title="Billiard table">billiard table</a>. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of <a href="/wiki/Reflection_(physics)" title="Reflection (physics)">reflections</a> at the boundary.</figcaption></figure> <a href="/wiki/Entropy" title="Entropy">Entropy</a> is broadly a measure of the disorder of some system. In <a href="/wiki/Statistical_thermodynamics" class="mw-redirect" title="Statistical thermodynamics">statistical thermodynamics</a>, the entropy S of some physical system is defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=-k\sum _{i}p_{i}\ln(p_{i}).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=-k\sum _{i}p_{i}\ln(p_{i}).\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4be67693caef12b846ed9cd173a0e7a340364d27" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.854ex; height:5.509ex;" alt="{\displaystyle S=-k\sum _{i}p_{i}\ln(p_{i}).\,}"> The sum is over all possible states i of the system in question, such as the positions of gas particles in a container. Moreover, pi is the probability that the state i is attained and k is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>. Similarly, <a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">entropy in information theory</a> measures the quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log2 N bits.<a href="#cite_note-91">[86]</a> <a href="/wiki/Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponents</a> use logarithms to gauge the degree of chaoticity of a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a>. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are <a href="/wiki/Chaos_theory" title="Chaos theory">chaotic</a> in a <a href="/wiki/Deterministic_system" title="Deterministic system">deterministic</a> way, because small measurement errors of the initial state predictably lead to largely different final states.<a href="#cite_note-92">[87]</a> At least one Lyapunov exponent of a deterministically chaotic system is positive. <div class="mw-heading mw-heading3"><h3 id="Fractals">Fractals</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=32" title="Edit section: Fractals">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sierpinski_dimension.svg" class="mw-file-description"><img alt="Parts of a triangle are removed in an iterated way." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Sierpinski_dimension.svg/400px-Sierpinski_dimension.svg.png" decoding="async" width="400" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Sierpinski_dimension.svg/600px-Sierpinski_dimension.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Sierpinski_dimension.svg/800px-Sierpinski_dimension.svg.png 2x" data-file-width="745" data-file-height="108" /></a><figcaption>The Sierpinski triangle (at the right) is constructed by repeatedly replacing <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangles</a> by three smaller ones.</figcaption></figure> Logarithms occur in definitions of the <a href="/wiki/Fractal_dimension" title="Fractal dimension">dimension</a> of <a href="/wiki/Fractal" title="Fractal">fractals</a>.<a href="#cite_note-93">[88]</a> Fractals are geometric objects that are self-similar in the sense that small parts reproduce, at least roughly, the entire global structure. The <a href="/wiki/Sierpinski_triangle" class="mw-redirect" title="Sierpinski triangle">Sierpinski triangle</a> (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the <a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a> of this structure ln(3)/ln(2) ≈ 1.58. Another logarithm-based notion of dimension is obtained by <a href="/wiki/Box-counting_dimension" class="mw-redirect" title="Box-counting dimension">counting the number of boxes</a> needed to cover the fractal in question. <div class="mw-heading mw-heading3"><h3 id="Music">Music</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=33" title="Edit section: Music">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:354px;max-width:354px"><div class="trow"><div class="tsingle" style="width:352px;max-width:352px"><div class="thumbimage"><a href="/wiki/File:4Octaves.and.Frequencies.svg" class="mw-file-description"><img alt="Four different octaves shown on a linear scale." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/4Octaves.and.Frequencies.svg/350px-4Octaves.and.Frequencies.svg.png" decoding="async" width="350" height="38" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/4Octaves.and.Frequencies.svg/525px-4Octaves.and.Frequencies.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/4Octaves.and.Frequencies.svg/700px-4Octaves.and.Frequencies.svg.png 2x" data-file-width="670" data-file-height="72" /></a></div></div></div><div class="trow"><div class="tsingle" style="width:352px;max-width:352px"><div class="thumbimage"><a href="/wiki/File:4Octaves.and.Frequencies.Ears.svg" class="mw-file-description"><img alt="Four different octaves shown on a logarithmic scale" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/4Octaves.and.Frequencies.Ears.svg/350px-4Octaves.and.Frequencies.Ears.svg.png" decoding="async" width="350" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/4Octaves.and.Frequencies.Ears.svg/525px-4Octaves.and.Frequencies.Ears.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/4Octaves.and.Frequencies.Ears.svg/700px-4Octaves.and.Frequencies.Ears.svg.png 2x" data-file-width="578" data-file-height="74" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Four different octaves shown on a linear scale, then shown on a logarithmic scale (as the ear hears them)</div></div></div></div> Logarithms are related to musical tones and <a href="/wiki/Interval_(music)" title="Interval (music)">intervals</a>. In <a href="/wiki/Equal_temperament" title="Equal temperament">equal temperament</a> tunings, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitch</a>, of the individual tones. In the <a href="/wiki/12-tone_equal_temperament" class="mw-redirect" title="12-tone equal temperament">12-tone equal temperament</a> tuning common in modern Western music, each <a href="/wiki/Octave" title="Octave">octave</a> (doubling of frequency) is broken into twelve equally spaced intervals called <a href="/wiki/Semitone" title="Semitone">semitones</a>. For example, if the <a href="/wiki/A_(musical_note)" title="A (musical note)">note A</a> has a frequency of 440 <a href="/wiki/Hertz" title="Hertz">Hz</a> then the note <a href="/wiki/B%E2%99%AD_(musical_note)" title="B♭ (musical note)">B-flat</a> has a frequency of 466 Hz. The interval between A and B-flat is a <a href="/wiki/Semitone" title="Semitone">semitone</a>, as is the one between B-flat and <a href="/wiki/B_(musical_note)" title="B (musical note)">B</a> (frequency 493 Hz). Accordingly, the frequency ratios agree: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {466}{440}}\approx {\frac {493}{466}}\approx 1.059\approx {\sqrt[{12}]{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>466</mn> <mn>440</mn> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>493</mn> <mn>466</mn> </mfrac> </mrow> <mo>≈</mo> <mn>1.059</mn> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mroot> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {466}{440}}\approx {\frac {493}{466}}\approx 1.059\approx {\sqrt[{12}]{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55acf246da64ba711e1717eb43ad81792220ab32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.993ex; height:5.343ex;" alt="{\displaystyle {\frac {466}{440}}\approx {\frac {493}{466}}\approx 1.059\approx {\sqrt[{12}]{2}}.}"> Intervals between arbitrary pitches can be measured in octaves by taking the base-2 logarithm of the <a href="/wiki/Frequency" title="Frequency">frequency</a> ratio, can be measured in equally tempered semitones by taking the base-21/12 logarithm (12 times the base-2 logarithm), or can be measured in <a href="/wiki/Cent_(music)" title="Cent (music)">cents</a>, hundredths of a semitone, by taking the base-21/1200 logarithm (1200 times the base-2 logarithm). The latter is used for finer encoding, as it is needed for finer measurements or non-equal temperaments.<a href="#cite_note-94">[89]</a> <table class="wikitable" style="text-align:center;"> <tbody><tr> <th>Interval (the two tones are played at the same time) </th> <td><a href="/wiki/72_tone_equal_temperament" class="mw-redirect" title="72 tone equal temperament">1/12 tone</a> <span data-nosnippet="" id="ooui-php-1" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/b\/b8\/1_step_in_72-et_on_C.mid\/1_step_in_72-et_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"1 step in 72-et on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/b/b8/1_step_in_72-et_on_C.mid/1_step_in_72-et_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:1_step_in_72-et_on_C.mid" title="File:1 step in 72-et on C.mid">ⓘ</a> </td> <td><a href="/wiki/Semitone" title="Semitone">Semitone</a> <span data-nosnippet="" id="ooui-php-2" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/8\/8a\/Minor_second_on_C.mid\/Minor_second_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Minor second on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8a/Minor_second_on_C.mid/Minor_second_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Minor_second_on_C.mid" title="File:Minor second on C.mid">ⓘ</a> </td> <td><a href="/wiki/Just_major_third" class="mw-redirect" title="Just major third">Just major third</a> <span data-nosnippet="" id="ooui-php-3" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/2\/2a\/Just_major_third_on_C.mid\/Just_major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Just major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/2/2a/Just_major_third_on_C.mid/Just_major_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Just_major_third_on_C.mid" title="File:Just major third on C.mid">ⓘ</a> </td> <td><a href="/wiki/Major_third" title="Major third">Major third</a> <span data-nosnippet="" id="ooui-php-4" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/9\/91\/Major_third_on_C.mid\/Major_third_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Major third on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/9/91/Major_third_on_C.mid/Major_third_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Major_third_on_C.mid" title="File:Major third on C.mid">ⓘ</a> </td> <td><a href="/wiki/Tritone" title="Tritone">Tritone</a> <span data-nosnippet="" id="ooui-php-5" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/5\/58\/Tritone_on_C.mid\/Tritone_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Tritone on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/5/58/Tritone_on_C.mid/Tritone_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Tritone_on_C.mid" title="File:Tritone on C.mid">ⓘ</a> </td> <td><a href="/wiki/Octave" title="Octave">Octave</a> <span data-nosnippet="" id="ooui-php-6" class="ext-phonos-PhonosButton noexcerpt oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-labelElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/f\/f0\/Perfect_octave_on_C.mid\/Perfect_octave_on_C.mid.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","label":{"html":"play"},"data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"Perfect octave on C.mid"},"classes":["ext-phonos-PhonosButton","noexcerpt"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f0/Perfect_octave_on_C.mid/Perfect_octave_on_C.mid.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button">play</a><a href="/wiki/File:Perfect_octave_on_C.mid" title="File:Perfect octave on C.mid">ⓘ</a> </td></tr> <tr> <th>Frequency ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {1}{72}}\approx 1.0097}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>72</mn> </mfrac> </mrow> </msup> <mo>≈</mo> <mn>1.0097</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {1}{72}}\approx 1.0097}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcce5a9aa9e216fc208a597f74d2d2b6248663e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.123ex; height:3.509ex;" alt="{\displaystyle 2^{\frac {1}{72}}\approx 1.0097}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {1}{12}}\approx 1.0595}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> </msup> <mo>≈</mo> <mn>1.0595</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {1}{12}}\approx 1.0595}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18664eeb9dbe129067dd89295c4928d54fda5f01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.123ex; height:3.509ex;" alt="{\displaystyle 2^{\frac {1}{12}}\approx 1.0595}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5}{4}}=1.25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>1.25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5}{4}}=1.25}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84da6d3ba8b361f151624067f68d609526e6e47b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.891ex; height:3.509ex;" alt="{\displaystyle {\tfrac {5}{4}}=1.25}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}2^{\frac {4}{12}}&={\sqrt[{3}]{2}}\\&\approx 1.2599\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>12</mn> </mfrac> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mn>1.2599</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2^{\frac {4}{12}}&={\sqrt[{3}]{2}}\\&\approx 1.2599\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76610ca7878ea438fa73bd50ac4df1fecce09b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:13.875ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}2^{\frac {4}{12}}&={\sqrt[{3}]{2}}\\&\approx 1.2599\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}2^{\frac {6}{12}}&={\sqrt {2}}\\&\approx 1.4142\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>12</mn> </mfrac> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mn>1.4142</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}2^{\frac {6}{12}}&={\sqrt {2}}\\&\approx 1.4142\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa178821ca7a1554106bf2244d08577f6f5d17fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:13.875ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}2^{\frac {6}{12}}&={\sqrt {2}}\\&\approx 1.4142\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\frac {12}{12}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>12</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{\frac {12}{12}}=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b7cc906bb2bc2787e32f7d1643b290d7b96766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.826ex; height:3.509ex;" alt="{\displaystyle 2^{\frac {12}{12}}=2}"> </td></tr> <tr> <th>Number of semitones <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12\log _{2}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12\log _{2}r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0bf873a44ac093c185619b4fc8e9852d569eaf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.174ex; height:2.676ex;" alt="{\displaystyle 12\log _{2}r}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc02e655226b1a0e18922e932efff50531c48eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{6}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx 3.8631}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> <mn>3.8631</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx 3.8631}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c58cbdc0681a74e9ebeb1b0ce4b25833bc75cee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.912ex; height:2.176ex;" alt="{\displaystyle \approx 3.8631}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a522d3aa5812a136a69f06e1b909d809e849be39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 12}"> </td></tr> <tr> <th>Number of cents <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1200\log _{2}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1200</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1200\log _{2}r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbcf502f2dd2fba6a0f6f6758ec0707c62484ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.499ex; height:2.676ex;" alt="{\displaystyle 1200\log _{2}r}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16{\tfrac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16{\tfrac {2}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14ffb45717ec74dc340583a93d6a788f6179382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.983ex; height:3.676ex;" alt="{\displaystyle 16{\tfrac {2}{3}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 100}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0572cd017c6d7936a12737c9d614a2f801f94a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 100}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \approx 386.31}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> <mn>386.31</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx 386.31}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0309fcfcbe7d98188d148b01f6f41d15ea416d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.912ex; height:2.176ex;" alt="{\displaystyle \approx 386.31}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 400}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>400</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 400}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8540670f7baa60a08a5dd4b12916c16fe6faf200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 400}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 600}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>600</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 600}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed5fbc94ba594303754ec8efd3d552547a93043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.487ex; height:2.176ex;" alt="{\displaystyle 600}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1200}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1200</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1200}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973054497debca94837d3a844349fe9221727dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.65ex; height:2.176ex;" alt="{\displaystyle 1200}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Number_theory">Number theory</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=34" title="Edit section: Number theory">edit</a>]</div> <a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithms</a> are closely linked to <a href="/wiki/Prime-counting_function" title="Prime-counting function">counting prime numbers</a> (2, 3, 5, 7, 11, ...), an important topic in <a href="/wiki/Number_theory" title="Number theory">number theory</a>. For any <a href="/wiki/Integer" title="Integer">integer</a> x, the quantity of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> less than or equal to x is denoted <a href="/wiki/Prime-counting_function" title="Prime-counting function">π(x)</a>. The <a href="/wiki/Prime_number_theorem" title="Prime number theorem">prime number theorem</a> asserts that π(x) is approximately given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{\ln(x)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{\ln(x)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7c35556de976b4896475a163d924bb9f3d83eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:6.561ex; height:5.509ex;" alt="{\displaystyle {\frac {x}{\ln(x)}},}"> in the sense that the ratio of π(x) and that fraction approaches 1 when x tends to infinity.<a href="#cite_note-95">[90]</a> As a consequence, the probability that a randomly chosen number between 1 and x is prime is inversely <a href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to the number of decimal digits of x. A far better estimate of π(x) is given by the <a href="/wiki/Logarithmic_integral_function" title="Logarithmic integral function">offset logarithmic integral</a> function Li(x), defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Li} (x)=\int _{2}^{x}{\frac {1}{\ln(t)}}\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Li} (x)=\int _{2}^{x}{\frac {1}{\ln(t)}}\,dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da577950b8b4c4ef726a3065afcdafa378dbc3fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.875ex; height:6.176ex;" alt="{\displaystyle \mathrm {Li} (x)=\int _{2}^{x}{\frac {1}{\ln(t)}}\,dt.}"> The <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, one of the oldest open mathematical <a href="/wiki/Conjecture" title="Conjecture">conjectures</a>, can be stated in terms of comparing π(x) and Li(x).<a href="#cite_note-96">[91]</a> The <a href="/wiki/Erd%C5%91s%E2%80%93Kac_theorem" title="Erdős–Kac theorem">Erdős–Kac theorem</a> describing the number of distinct <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> also involves the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a>. The logarithm of n <a href="/wiki/Factorial" title="Factorial">factorial</a>, n! = 1 · 2 · ... · n, is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(n!)=\ln(1)+\ln(2)+\cdots +\ln(n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(n!)=\ln(1)+\ln(2)+\cdots +\ln(n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/207af1498eca9a74c5e19ecab897d915d1052c95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.746ex; height:2.843ex;" alt="{\displaystyle \ln(n!)=\ln(1)+\ln(2)+\cdots +\ln(n).}"> This can be used to obtain <a href="/wiki/Stirling%27s_formula" class="mw-redirect" title="Stirling's formula">Stirling's formula</a>, an approximation of n! for large n.<a href="#cite_note-97">[92]</a> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=35" title="Edit section: Generalizations">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Complex_logarithm">Complex logarithm</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=36" title="Edit section: Complex logarithm">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Complex_logarithm" title="Complex logarithm">Complex logarithm</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_number_illustration_multiple_arguments.svg" class="mw-file-description"><img alt="An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x-axis." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Complex_number_illustration_multiple_arguments.svg/220px-Complex_number_illustration_multiple_arguments.svg.png" decoding="async" width="220" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Complex_number_illustration_multiple_arguments.svg/330px-Complex_number_illustration_multiple_arguments.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Complex_number_illustration_multiple_arguments.svg/440px-Complex_number_illustration_multiple_arguments.svg.png 2x" data-file-width="204" data-file-height="217" /></a><figcaption>Polar form of z = x + iy. Both φ and φ' are arguments of z.</figcaption></figure> All the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> a that solve the equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{a}=z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>=</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{a}=z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc3322ef38b06276c3bee59d656769b6edf531f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.372ex; height:2.343ex;" alt="{\displaystyle e^{a}=z}"> are called complex logarithms of z, when z is (considered as) a complex number. A complex number is commonly represented as z = x + iy, where x and y are real numbers and i is an <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>, the square of which is −1. Such a number can be visualized by a point in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, as shown at the right. The <a href="/wiki/Polar_form" class="mw-redirect" title="Polar form">polar form</a> encodes a non-zero complex number z by its <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a>, that is, the (positive, real) distance r to the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>, and an angle between the real (x) axis Re and the line passing through both the origin and z. This angle is called the <a href="/wiki/Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> of z. The absolute value r of z is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r={\sqrt {x^{2}+y^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle r={\sqrt {x^{2}+y^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c0606ea41983c11ed8e73fc1507ce58ad316ef0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.557ex; height:3.509ex;" alt="{\displaystyle \textstyle r={\sqrt {x^{2}+y^{2}}}.}"> Using the geometrical interpretation of <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> and their periodicity in 2π, any complex number z may be denoted as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}z&=x+iy\\&=r(\cos \varphi +i\sin \varphi )\\&=r(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>i</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}z&=x+iy\\&=r(\cos \varphi +i\sin \varphi )\\&=r(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )),\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6851b66ce18570803cf1ffdc5b789016559b218" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:38.191ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}z&=x+iy\\&=r(\cos \varphi +i\sin \varphi )\\&=r(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )),\end{aligned}}}"> for any integer number k. Evidently the argument of z is not uniquely specified: both φ and φ' = φ + 2kπ are valid arguments of z for all integers k, because adding 2kπ <a href="/wiki/Radian" title="Radian">radians</a> or k⋅360°<a href="#cite_note-98">[nb 6]</a> to φ corresponds to "winding" around the origin counter-clock-wise by k <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turns</a>. The resulting complex number is always z, as illustrated at the right for k = 1. One may select exactly one of the possible arguments of z as the so-called principal argument, denoted Arg(z), with a capital A, by requiring φ to belong to one, conveniently selected turn, e.g. −π < φ ≤ π<a href="#cite_note-99">[93]</a> or 0 ≤ φ < 2π.<a href="#cite_note-100">[94]</a> These regions, where the argument of z is uniquely determined are called <a href="/wiki/Principal_branch" title="Principal branch">branches</a> of the argument function. <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Complex_log_domain.svg" class="mw-file-description"><img alt="A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_log_domain.svg/220px-Complex_log_domain.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_log_domain.svg/330px-Complex_log_domain.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_log_domain.svg/440px-Complex_log_domain.svg.png 2x" data-file-width="569" data-file-height="426" /></a><figcaption>The principal branch (-π, π) of the complex logarithm, Log(z). The black point at z = 1 corresponds to absolute value zero and brighter colors refer to bigger absolute values. The <a href="/wiki/Hue" title="Hue">hue</a> of the color encodes the argument of Log(z).</figcaption></figure> <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a> connects the <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> and <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> to the <a href="/wiki/Complex_exponential" class="mw-redirect" title="Complex exponential">complex exponential</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>φ</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/396158ab1664889849843ce26a324ed8dbbf841e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.515ex; height:3.176ex;" alt="{\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi .}"> Using this formula, and again the periodicity, the following identities hold:<a href="#cite_note-101">[95]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}z&=r\left(\cos \varphi +i\sin \varphi \right)\\&=r\left(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )\right)\\&=re^{i(\varphi +2k\pi )}\\&=e^{\ln(r)}e^{i(\varphi +2k\pi )}\\&=e^{\ln(r)+i(\varphi +2k\pi )}=e^{a_{k}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}z&=r\left(\cos \varphi +i\sin \varphi \right)\\&=r\left(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )\right)\\&=re^{i(\varphi +2k\pi )}\\&=e^{\ln(r)}e^{i(\varphi +2k\pi )}\\&=e^{\ln(r)+i(\varphi +2k\pi )}=e^{a_{k}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7f91b7eb61729d49185f2c48d2eb32c40a11a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:37.931ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}z&=r\left(\cos \varphi +i\sin \varphi \right)\\&=r\left(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )\right)\\&=re^{i(\varphi +2k\pi )}\\&=e^{\ln(r)}e^{i(\varphi +2k\pi )}\\&=e^{\ln(r)+i(\varphi +2k\pi )}=e^{a_{k}},\end{aligned}}}"> where ln(r) is the unique real natural logarithm, ak denote the complex logarithms of z, and k is an arbitrary integer. Therefore, the complex logarithms of z, which are all those complex values ak for which the ak-th power of e equals z, are the infinitely many values <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}=\ln(r)+i(\varphi +2k\pi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k}=\ln(r)+i(\varphi +2k\pi ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22659f325d07ee703329595fc7cf9d372c4fb086" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.38ex; height:2.843ex;" alt="{\displaystyle a_{k}=\ln(r)+i(\varphi +2k\pi ),}"> for arbitrary integers k. Taking k such that φ + 2kπ is within the defined interval for the principal arguments, then ak is called the principal value of the logarithm, denoted Log(z), again with a capital L. The principal argument of any positive real number x is 0; hence Log(x) is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers <a href="/wiki/Exponentiation#Failure_of_power_and_logarithm_identities" title="Exponentiation">do not generalize</a> to the principal value of the complex logarithm.<a href="#cite_note-102">[96]</a> The illustration at the right depicts Log(z), confining the arguments of z to the interval (−π, π]. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real x axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding k-value of the continuously neighboring branch. Such a locus is called a <a href="/wiki/Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a>. Dropping the range restrictions on the argument makes the relations "argument of z", and consequently the "logarithm of z", <a href="/wiki/Multi-valued_function" class="mw-redirect" title="Multi-valued function">multi-valued functions</a>. <div class="mw-heading mw-heading3"><h3 id="Inverses_of_other_exponential_functions">Inverses of other exponential functions</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=37" title="Edit section: Inverses of other exponential functions">edit</a>]</div> Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the <a href="/wiki/Logarithm_of_a_matrix" title="Logarithm of a matrix">logarithm of a matrix</a> is the (multi-valued) inverse function of the <a href="/wiki/Matrix_exponential" title="Matrix exponential">matrix exponential</a>.<a href="#cite_note-103">[97]</a> Another example is the <a href="/wiki/P-adic_logarithm_function" class="mw-redirect" title="P-adic logarithm function">p-adic logarithm</a>, the inverse function of the <a href="/wiki/P-adic_exponential_function" title="P-adic exponential function">p-adic exponential</a>. Both are defined via Taylor series analogous to the real case.<a href="#cite_note-104">[98]</a> In the context of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the <a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">exponential map</a> maps the <a href="/wiki/Tangent_space" title="Tangent space">tangent space</a> at a point of a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">manifold</a> to a <a href="/wiki/Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> of that point. Its inverse is also called the logarithmic (or log) map.<a href="#cite_note-105">[99]</a> In the context of <a href="/wiki/Finite_group" title="Finite group">finite groups</a> exponentiation is given by repeatedly multiplying one group element b with itself. The <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a> is the integer n solving the equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}=x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{n}=x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f90e5e0851759d0490e085cf1888338465384143" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.291ex; height:2.676ex;" alt="{\displaystyle b^{n}=x,}"> where x is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in <a href="/wiki/Public_key_cryptography" class="mw-redirect" title="Public key cryptography">public key cryptography</a>, such as for example in the <a href="/wiki/Diffie%E2%80%93Hellman_key_exchange" title="Diffie–Hellman key exchange">Diffie–Hellman key exchange</a>, a routine that allows secure exchanges of <a href="/wiki/Cryptography" title="Cryptography">cryptographic</a> keys over unsecured information channels.<a href="#cite_note-106">[100]</a> <a href="/wiki/Zech%27s_logarithm" title="Zech's logarithm">Zech's logarithm</a> is related to the discrete logarithm in the multiplicative group of non-zero elements of a <a href="/wiki/Finite_field" title="Finite field">finite field</a>.<a href="#cite_note-107">[101]</a> Further logarithm-like inverse functions include the double logarithm ln(ln(x)), the <a href="/wiki/Super-logarithm" class="mw-redirect" title="Super-logarithm">super- or hyper-4-logarithm</a> (a slight variation of which is called <a href="/wiki/Iterated_logarithm" title="Iterated logarithm">iterated logarithm</a> in computer science), the <a href="/wiki/Lambert_W_function" title="Lambert W function">Lambert W function</a>, and the <a href="/wiki/Logit" title="Logit">logit</a>. They are the inverse functions of the <a href="/wiki/Double_exponential_function" title="Double exponential function">double exponential function</a>, <a href="/wiki/Tetration" title="Tetration">tetration</a>, of f(w) = wew,<a href="#cite_note-108">[102]</a> and of the <a href="/wiki/Logistic_function" title="Logistic function">logistic function</a>, respectively.<a href="#cite_note-109">[103]</a> <div class="mw-heading mw-heading3"><h3 id="Related_concepts">Related concepts</h3>[<a href="/w/index.php?title=Logarithm&action=edit&section=38" title="Edit section: Related concepts">edit</a>]</div> From the perspective of <a href="/wiki/Group_theory" title="Group theory">group theory</a>, the identity log(cd) = log(c) + log(d) expresses a <a href="/wiki/Group_isomorphism" title="Group isomorphism">group isomorphism</a> between positive <a href="/wiki/Real_number" title="Real number">reals</a> under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.<a href="#cite_note-110">[104]</a> By means of that isomorphism, the <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a> (<a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>) dx on the reals corresponds to the Haar measure dx/x on the positive reals.<a href="#cite_note-111">[105]</a> The non-negative reals not only have a multiplication, but also have addition, and form a <a href="/wiki/Semiring" title="Semiring">semiring</a>, called the <a href="/wiki/Probability_semiring" class="mw-redirect" title="Probability semiring">probability semiring</a>; this is in fact a <a href="/wiki/Semifield" title="Semifield">semifield</a>. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (<a href="/wiki/LogSumExp" title="LogSumExp">LogSumExp</a>), giving an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> of semirings between the probability semiring and the <a href="/wiki/Log_semiring" title="Log semiring">log semiring</a>. <a href="/wiki/Logarithmic_form" title="Logarithmic form">Logarithmic one-forms </a>df/f appear in <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a> and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> as <a href="/wiki/Differential_form" title="Differential form">differential forms</a> with logarithmic <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a>.<a href="#cite_note-112">[106]</a> The <a href="/wiki/Polylogarithm" title="Polylogarithm">polylogarithm</a> is the function defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Li</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c16a216f9168ba23df2d07ceb32c6929a70c4e1b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.538ex; height:6.843ex;" alt="{\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.}"> It is related to the <a href="/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a> by Li1 (z) = −ln(1 − z). Moreover, Lis (1) equals the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> ζ(s).<a href="#cite_note-113">[107]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=39" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 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src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Nuvola_apps_kcmsystem.svg/28px-Nuvola_apps_kcmsystem.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Nuvola_apps_kcmsystem.svg/42px-Nuvola_apps_kcmsystem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Nuvola_apps_kcmsystem.svg/56px-Nuvola_apps_kcmsystem.svg.png 2x" data-file-width="128" data-file-height="128" /></a><a href="/wiki/Portal:Engineering" title="Portal:Engineering">Engineering portal</a></li></ul> <ul><li><a href="/wiki/Decimal_exponent" class="mw-redirect" title="Decimal exponent">Decimal exponent</a> (dex)</li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Index_of_logarithm_articles" title="Index of logarithm articles">Index of logarithm articles</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=40" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-3"><a href="#cite_ref-3">^</a> The restrictions on x and b are explained in the section <a href="#Analytic_properties">"Analytic properties"</a>. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> Proof: Taking the logarithm to base k of the defining identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x=b^{\log _{b}x},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=b^{\log _{b}x},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/361b01fa020a096dc96de0f7b07de6eadba74597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.47ex; height:2.843ex;" alt="{\textstyle x=b^{\log _{b}x},}"> one gets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{k}x=\log _{k}\left(b^{\log _{b}x}\right)=\log _{b}x\cdot \log _{k}b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{k}x=\log _{k}\left(b^{\log _{b}x}\right)=\log _{b}x\cdot \log _{k}b.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9963d4106a92dfa004d0110b8415407fa52ce9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.957ex; height:3.343ex;" alt="{\displaystyle \log _{k}x=\log _{k}\left(b^{\log _{b}x}\right)=\log _{b}x\cdot \log _{k}b.}"> The formula follows by solving for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f58e6c48d751c89792f8c702a8a2537f1621d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.273ex; height:2.676ex;" alt="{\displaystyle \log _{b}x.}"> </li> <li id="cite_note-adaa-21"><a href="#cite_ref-adaa_21-0">^</a> z Some mathematicians disapprove of this notation. In his 1985 autobiography, <a href="/wiki/Paul_Halmos" title="Paul Halmos">Paul Halmos</a> criticized what he considered the "childish ln notation", which he said no mathematician had ever used.<a href="#cite_note-18">[16]</a> The notation was invented by the 19th century mathematician <a href="/wiki/Irving_Stringham" title="Irving Stringham">I. Stringham</a>.<a href="#cite_note-19">[17]</a><a href="#cite_note-20">[18]</a> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> The same series holds for the principal value of the complex logarithm for complex numbers z satisfying |z − 1| < 1. </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> The same series holds for the principal value of the complex logarithm for complex numbers z with positive real part. </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> See <a href="/wiki/Radian" title="Radian">radian</a> for the conversion between 2<a href="/wiki/Pi" title="Pi">π</a> and 360 <a href="/wiki/Degree_(angle)" title="Degree (angle)">degree</a>. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=41" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHobson1914" class="citation cs2">Hobson, Ernest William (1914), <a rel="nofollow" class="external text" href="http://archive.org/details/johnnapierinvent00hobsiala">John Napier and the invention of logarithms, 1614; a lecture</a>, University of California Libraries, Cambridge : University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=John+Napier+and+the+invention+of+logarithms%2C+1614%3B+a+lecture&rft.pub=Cambridge+%3A+University+Press&rft.date=1914&rft.aulast=Hobson&rft.aufirst=Ernest+William&rft_id=http%3A%2F%2Farchive.org%2Fdetails%2Fjohnnapierinvent00hobsiala&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRemmert,_Reinhold.1991" class="citation cs2">Remmert, Reinhold. (1991), Theory of complex functions, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0387971955" title="Special:BookSources/0387971955"><bdi>0387971955</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21118309">21118309</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+complex+functions&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1991&rft_id=info%3Aoclcnum%2F21118309&rft.isbn=0387971955&rft.au=Remmert%2C+Reinhold.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKateBhapkar2009" class="citation cs2">Kate, S.K.; Bhapkar, H.R. (2009), Basics Of Mathematics, Pune: Technical Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-8431-755-8" title="Special:BookSources/978-81-8431-755-8"><bdi>978-81-8431-755-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basics+Of+Mathematics&rft.place=Pune&rft.pub=Technical+Publications&rft.date=2009&rft.isbn=978-81-8431-755-8&rft.aulast=Kate&rft.aufirst=S.K.&rft.au=Bhapkar%2C+H.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, chapter 1 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> All statements in this section can be found in Douglas Downing <a href="#CITEREFDowning2003">2003</a>, p. 275 or Kate & Bhapkar <a href="#CITEREFKateBhapkar2009">2009</a>, p. 1-1, for example. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernsteinBernstein1999" class="citation cs2">Bernstein, Stephen; Bernstein, Ruth (1999), <a rel="nofollow" class="external text" href="https://archive.org/details/schaumsoutlineof00bern">Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probability</a>, Schaum's outline series, New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-005023-5" title="Special:BookSources/978-0-07-005023-5"><bdi>978-0-07-005023-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+outline+of+theory+and+problems+of+elements+of+statistics.+I%2C+Descriptive+statistics+and+probability&rft.place=New+York&rft.series=Schaum%27s+outline+series&rft.pub=McGraw-Hill&rft.date=1999&rft.isbn=978-0-07-005023-5&rft.aulast=Bernstein&rft.aufirst=Stephen&rft.au=Bernstein%2C+Ruth&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fschaumsoutlineof00bern&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, p. 21 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDowning2003" class="citation book cs2">Downing, Douglas (2003), <a rel="nofollow" class="external text" href="https://archive.org/details/algebraeasyway00down_0">Algebra the Easy Way</a>, Barron's Educational Series, Hauppauge, NY: Barron's, chapter 17, p. 275, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7641-1972-9" title="Special:BookSources/978-0-7641-1972-9"><bdi>978-0-7641-1972-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+the+Easy+Way&rft.place=Hauppauge%2C+NY&rft.series=Barron%27s+Educational+Series&rft.pages=chapter-17%2C+p.-275&rft.pub=Barron%27s&rft.date=2003&rft.isbn=978-0-7641-1972-9&rft.aulast=Downing&rft.aufirst=Douglas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebraeasyway00down_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWegener2005" class="citation book cs2">Wegener, Ingo (2005), Complexity Theory: Exploring the limits of efficient algorithms, Berlin, DE / New York, NY: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, p. 20, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-21045-0" title="Special:BookSources/978-3-540-21045-0"><bdi>978-3-540-21045-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complexity+Theory%3A+Exploring+the+limits+of+efficient+algorithms&rft.place=Berlin%2C+DE+%2F+New+York%2C+NY&rft.pages=20&rft.pub=Springer-Verlag&rft.date=2005&rft.isbn=978-3-540-21045-0&rft.aulast=Wegener&rft.aufirst=Ingo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_Lubbe1997" class="citation book cs2">van der Lubbe, Jan C.A. (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tBuI_6MQTcwC&pg=PA3">Information Theory</a>, Cambridge University Press, p. 3, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-46760-5" title="Special:BookSources/978-0-521-46760-5"><bdi>978-0-521-46760-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Theory&rft.pages=3&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=978-0-521-46760-5&rft.aulast=van+der+Lubbe&rft.aufirst=Jan+C.A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtBuI_6MQTcwC%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAllenTriantaphillidou2011" class="citation book cs2">Allen, Elizabeth; Triantaphillidou, Sophie (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IfWivY3mIgAC&pg=PA228">The Manual of Photography</a>, Taylor & Francis, p. 228, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-240-52037-7" title="Special:BookSources/978-0-240-52037-7"><bdi>978-0-240-52037-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Manual+of+Photography&rft.pages=228&rft.pub=Taylor+%26+Francis&rft.date=2011&rft.isbn=978-0-240-52037-7&rft.aulast=Allen&rft.aufirst=Elizabeth&rft.au=Triantaphillidou%2C+Sophie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIfWivY3mIgAC%26pg%3DPA228&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFParkhurst2007" class="citation book cs2">Parkhurst, David F. (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h6yq_lOr8Z4C&pg=PA288">Introduction to Applied Mathematics for Environmental Science</a> (illustrated ed.), Springer Science & Business Media, p. 288, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-34228-3" title="Special:BookSources/978-0-387-34228-3"><bdi>978-0-387-34228-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Applied+Mathematics+for+Environmental+Science&rft.pages=288&rft.edition=illustrated&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-0-387-34228-3&rft.aulast=Parkhurst&rft.aufirst=David+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dh6yq_lOr8Z4C%26pg%3DPA288&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodrichTamassia2002" class="citation book cs2"><a href="/wiki/Michael_T._Goodrich" title="Michael T. Goodrich">Goodrich, Michael T.</a>; <a href="/wiki/Roberto_Tamassia" title="Roberto Tamassia">Tamassia, Roberto</a> (2002), Algorithm Design: Foundations, analysis, and internet examples, John Wiley & Sons, p. 23, <q>One of the interesting and sometimes even surprising aspects of the analysis of data structures and algorithms is the ubiquitous presence of logarithms ... As is the custom in the computing literature, we omit writing the base b of the logarithm when b = 2 .</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithm+Design%3A+Foundations%2C+analysis%2C+and+internet+examples&rft.pages=23&rft.pub=John+Wiley+%26+Sons&rft.date=2002&rft.aulast=Goodrich&rft.aufirst=Michael+T.&rft.au=Tamassia%2C+Roberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1984" class="citation book cs2">Rudin, Walter (1984), "Theorem 3.29", <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi">Principles of Mathematical Analysis</a> (3rd ed., International student ed.), Auckland, NZ: McGraw-Hill International, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-085613-4" title="Special:BookSources/978-0-07-085613-4"><bdi>978-0-07-085613-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Theorem+3.29&rft.btitle=Principles+of+Mathematical+Analysis&rft.place=Auckland%2C+NZ&rft.edition=3rd+ed.%2C+International+student&rft.pub=McGraw-Hill+International&rft.date=1984&rft.isbn=978-0-07-085613-4&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesofmath00rudi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation report cs2">"Part 2: Mathematics", [title not cited], Quantities and units (Report), <a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">International Organization for Standardization</a>, 2019, <a href="/wiki/ISO_80000-2" class="mw-redirect" title="ISO 80000-2">ISO 80000-2</a>:2019 / EN <a href="/wiki/ISO_80000-2" class="mw-redirect" title="ISO 80000-2">ISO 80000-2</a> </cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=%3Cspan+style%3D%22color%3Agray%22%3E%5Btitle+not+cited%5D%3C%2Fspan%3E&rft.pub=International+Organization+for+Standardization&rft.date=2019&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> <dl><dd></dd></dl> See also <a href="/wiki/ISO_80000-2" class="mw-redirect" title="ISO 80000-2">ISO 80000-2</a> . </li> <li id="cite_note-gullberg-16"><a href="#cite_ref-gullberg_16-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGullberg1997" class="citation book cs2">Gullberg, Jan (1997), <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsfromb1997gull">Mathematics: From the birth of numbers</a>, New York, NY: W.W. Norton & Co, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-393-04002-9" title="Special:BookSources/978-0-393-04002-9"><bdi>978-0-393-04002-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%3A+From+the+birth+of+numbers&rft.place=New+York%2C+NY&rft.pub=W.W.+Norton+%26+Co&rft.date=1997&rft.isbn=978-0-393-04002-9&rft.aulast=Gullberg&rft.aufirst=Jan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsfromb1997gull&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a href="/wiki/The_Chicago_Manual_of_Style" title="The Chicago Manual of Style">The Chicago Manual of Style</a> (25th ed.), University of Chicago Press, 2003, p. 530</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Chicago+Manual+of+Style&rft.pages=530&rft.edition=25th&rft.pub=University+of+Chicago+Press&rft.date=2003&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">. </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1985" class="citation book cs2"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, P.</a> (1985), I Want to be a Mathematician: An automathography, Berlin, DE / New York, NY: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-96078-4" title="Special:BookSources/978-0-387-96078-4"><bdi>978-0-387-96078-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=I+Want+to+be+a+Mathematician%3A+An+automathography&rft.place=Berlin%2C+DE+%2F+New+York%2C+NY&rft.pub=Springer-Verlag&rft.date=1985&rft.isbn=978-0-387-96078-4&rft.aulast=Halmos&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStringham1893" class="citation book cs2"><a href="/wiki/Irving_Stringham" title="Irving Stringham">Stringham, I.</a> (1893), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hPEKAQAAIAAJ&pg=PA13">Uniplanar Algebra</a>, The Berkeley Press, p. xiii, <q>Being part I of a propædeutic to the higher mathematical analysis</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Uniplanar+Algebra&rft.pages=%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3Exiii%3C%2Fspan%3E&rft.pub=The+Berkeley+Press&rft.date=1893&rft.aulast=Stringham&rft.aufirst=I.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhPEKAQAAIAAJ%26pg%3DPA13&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreedman2006" class="citation book cs2">Freedman, Roy S. (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=APJ7QeR_XPkC&pg=PA5">Introduction to Financial Technology</a>, Amsterdam: Academic Press, p. 59, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-370478-8" title="Special:BookSources/978-0-12-370478-8"><bdi>978-0-12-370478-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Financial+Technology&rft.place=Amsterdam&rft.pages=59&rft.pub=Academic+Press&rft.date=2006&rft.isbn=978-0-12-370478-8&rft.aulast=Freedman&rft.aufirst=Roy+S.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAPJ7QeR_XPkC%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNapier1614" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/John_Napier" title="John Napier">Napier, John</a> (1614), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN527914568&DMDID=DMDLOG_0001&LOGID=LOG_0001&PHYSID=PHYS_0001">Mirifici Logarithmorum Canonis Descriptio</a> [The Description of the Wonderful Canon of Logarithms] (in Latin), Edinburgh, Scotland: Andrew Hart</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mirifici+Logarithmorum+Canonis+Descriptio&rft.place=Edinburgh%2C+Scotland&rft.pub=Andrew+Hart&rft.date=1614&rft.aulast=Napier&rft.aufirst=John&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN527914568%26DMDID%3DDMDLOG_0001%26LOGID%3DLOG_0001%26PHYSID%3DPHYS_0001&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> <div class="paragraphbreak" style="margin-top:0.5em"></div> The sequel ... Constructio was published posthumously: <div class="paragraphbreak" style="margin-top:0.5em"></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNapier1619" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/John_Napier" title="John Napier">Napier, John</a> (1619), Mirifici Logarithmorum Canonis Constructio [The Construction of the Wonderful Rule of Logarithms] (in Latin), Edinburgh: Andrew Hart</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mirifici+Logarithmorum+Canonis+Constructio&rft.place=Edinburgh&rft.pub=Andrew+Hart&rft.date=1619&rft.aulast=Napier&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> <div class="paragraphbreak" style="margin-top:0.5em"></div> Ian Bruce has made an <a rel="nofollow" class="external text" href="http://17centurymaths.com/contents/napiercontents.html">annotated translation of both books</a> (2012), available from 17centurymaths.com. </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHobson1914" class="citation cs2">Hobson, Ernest William (1914), <a rel="nofollow" class="external text" href="https://archive.org/details/johnnapierinvent00hobsiala">John Napier and the invention of logarithms, 1614</a>, Cambridge: The University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=John+Napier+and+the+invention+of+logarithms%2C+1614&rft.place=Cambridge&rft.pub=The+University+Press&rft.date=1914&rft.aulast=Hobson&rft.aufirst=Ernest+William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fjohnnapierinvent00hobsiala&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-folkerts-24"><a href="#cite_ref-folkerts_24-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFolkertsLaunertThom2016" class="citation cs2">Folkerts, Menso; Launert, Dieter; Thom, Andreas (2016), "Jost Bürgi's method for calculating sines", <a href="/wiki/Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a>, 43 (2): 133–147, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1510.03180">1510.03180</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.hm.2016.03.001">10.1016/j.hm.2016.03.001</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3489006">3489006</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119326088">119326088</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=Jost+B%C3%BCrgi%27s+method+for+calculating+sines&rft.volume=43&rft.issue=2&rft.pages=133-147&rft.date=2016&rft_id=info%3Aarxiv%2F1510.03180&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3489006%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119326088%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.hm.2016.03.001&rft.aulast=Folkerts&rft.aufirst=Menso&rft.au=Launert%2C+Dieter&rft.au=Thom%2C+Andreas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Burgi.html">"Jost Bürgi (1552 – 1632)"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Jost+B%C3%BCrgi+%281552+%E2%80%93+1632%29&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FBurgi.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> William Gardner (1742) Tables of Logarithms </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPierce1977" class="citation cs2">Pierce, R. C. Jr. (January 1977), "A brief history of logarithms", <a href="/wiki/The_Two-Year_College_Mathematics_Journal" class="mw-redirect" title="The Two-Year College Mathematics Journal">The Two-Year College Mathematics Journal</a>, 8 (1): 22–26, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3026878">10.2307/3026878</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3026878">3026878</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Two-Year+College+Mathematics+Journal&rft.atitle=A+brief+history+of+logarithms&rft.volume=8&rft.issue=1&rft.pages=22-26&rft.date=1977-01&rft_id=info%3Adoi%2F10.2307%2F3026878&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3026878%23id-name%3DJSTOR&rft.aulast=Pierce&rft.aufirst=R.+C.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> Enrique Gonzales-Velasco (2011) Journey through Mathematics – Creative Episodes in its History, §2.4 Hyperbolic logarithms, p. 117, Springer <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-92153-2" title="Special:BookSources/978-0-387-92153-2">978-0-387-92153-2</a> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <a href="/wiki/Florian_Cajori" title="Florian Cajori">Florian Cajori</a> (1913) "History of the exponential and logarithm concepts", <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a> 20: 5, 35, 75, 107, 148, 173, 205 </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillwell2010" class="citation cs2">Stillwell, J. (2010), Mathematics and Its History (3rd ed.), Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Its+History&rft.edition=3rd&rft.pub=Springer&rft.date=2010&rft.aulast=Stillwell&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBryant1907" class="citation cs2">Bryant, Walter W. (1907), <a rel="nofollow" class="external text" href="https://archive.org/stream/ahistoryastrono01bryagoog#page/n72/mode/2up">A History of Astronomy</a>, London: Methuen & Co</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Astronomy&rft.place=London&rft.pub=Methuen+%26+Co&rft.date=1907&rft.aulast=Bryant&rft.aufirst=Walter+W.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fahistoryastrono01bryagoog%23page%2Fn72%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, p. 44 </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbramowitzStegun1972" class="citation cs2"><a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="/wiki/Irene_Stegun" title="Irene Stegun">Stegun, Irene A.</a>, eds. (1972), <a href="/wiki/Handbook_of_Mathematical_Functions_with_Formulas,_Graphs,_and_Mathematical_Tables" class="mw-redirect" title="Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a> (10th ed.), New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61272-0" title="Special:BookSources/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=New+York&rft.edition=10th&rft.pub=Dover+Publications&rft.date=1972&rft.isbn=978-0-486-61272-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 4.7., p. 89 </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCampbell-Kelly2003" class="citation cs2">Campbell-Kelly, Martin (2003), <a href="/wiki/The_History_of_Mathematical_Tables" title="The History of Mathematical Tables">The history of mathematical tables: from Sumer to spreadsheets</a>, Oxford scholarship online, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850841-0" title="Special:BookSources/978-0-19-850841-0"><bdi>978-0-19-850841-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+history+of+mathematical+tables%3A+from+Sumer+to+spreadsheets&rft.series=Oxford+scholarship+online&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=978-0-19-850841-0&rft.aulast=Campbell-Kelly&rft.aufirst=Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 2 </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpiegelMoyer2006" class="citation cs2">Spiegel, Murray R.; Moyer, R.E. (2006), Schaum's outline of college algebra, Schaum's outline series, New York: <a href="/wiki/McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-145227-4" title="Special:BookSources/978-0-07-145227-4"><bdi>978-0-07-145227-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+outline+of+college+algebra&rft.place=New+York&rft.series=Schaum%27s+outline+series&rft.pub=McGraw-Hill&rft.date=2006&rft.isbn=978-0-07-145227-4&rft.aulast=Spiegel&rft.aufirst=Murray+R.&rft.au=Moyer%2C+R.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, p. 264 </li> <li id="cite_note-ReferenceA-35"><a href="#cite_ref-ReferenceA_35-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaor2009" class="citation cs2">Maor, Eli (2009), E: The Story of a Number, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, sections 1, 13, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14134-3" title="Special:BookSources/978-0-691-14134-3"><bdi>978-0-691-14134-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=E%3A+The+Story+of+a+Number&rft.pages=sections+1%2C+13&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-14134-3&rft.aulast=Maor&rft.aufirst=Eli&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevlin2004" class="citation cs2"><a href="/wiki/Keith_Devlin" title="Keith Devlin">Devlin, Keith</a> (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uQHF7bcm4k4C">Sets, functions, and logic: an introduction to abstract mathematics</a>, Chapman & Hall/CRC mathematics (3rd ed.), Boca Raton, Fla: Chapman & Hall/CRC, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-449-1" title="Special:BookSources/978-1-58488-449-1"><bdi>978-1-58488-449-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets%2C+functions%2C+and+logic%3A+an+introduction+to+abstract+mathematics&rft.place=Boca+Raton%2C+Fla&rft.series=Chapman+%26+Hall%2FCRC+mathematics&rft.edition=3rd&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2004&rft.isbn=978-1-58488-449-1&rft.aulast=Devlin&rft.aufirst=Keith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuQHF7bcm4k4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, or see the references in <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> </li> <li id="cite_note-LangIII.3-37">^ <a href="#cite_ref-LangIII.3_37-0">a</a> <a href="#cite_ref-LangIII.3_37-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1997" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1997), Undergraduate analysis, <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a> (2nd ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-2698-5">10.1007/978-1-4757-2698-5</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94841-6" title="Special:BookSources/978-0-387-94841-6"><bdi>978-0-387-94841-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1476913">1476913</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Undergraduate+analysis&rft.place=Berlin%2C+New+York&rft.series=Undergraduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1997&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1476913%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-2698-5&rft.isbn=978-0-387-94841-6&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section III.3 </li> <li id="cite_note-LangIV.2-38">^ <a href="#cite_ref-LangIV.2_38-0">a</a> <a href="#cite_ref-LangIV.2_38-1">b</a> Lang <a href="#CITEREFLang1997">1997</a>, section IV.2 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDieudonné1969" class="citation cs2">Dieudonné, Jean (1969), Foundations of Modern Analysis, vol. 1, Academic Press, p. 84</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Modern+Analysis&rft.pages=84&rft.pub=Academic+Press&rft.date=1969&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> item (4.3.1) </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=d/dx(Log(b,x))">"Calculation of d/dx(Log(b,x))"</a>, Wolfram Alpha, <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>, retrieved 15 March 2011</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wolfram+Alpha&rft.atitle=Calculation+of+d%2Fdx%28Log%28b%2Cx%29%29&rft_id=http%3A%2F%2Fwww.wolframalpha.com%2Finput%2F%3Fi%3Dd%2Fdx%28Log%28b%2Cx%29%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1998" class="citation cs2"><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a> (1998), Calculus: an intuitive and physical approach, Dover books on mathematics, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-40453-0" title="Special:BookSources/978-0-486-40453-0"><bdi>978-0-486-40453-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Calculus%3A+an+intuitive+and+physical+approach&rft.place=New+York&rft.series=Dover+books+on+mathematics&rft.pub=Dover+Publications&rft.date=1998&rft.isbn=978-0-486-40453-0&rft.aulast=Kline&rft.aufirst=Morris&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, p. 386 </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=Integrate(ln(x))">"Calculation of Integrate(ln(x))"</a>, Wolfram Alpha, Wolfram Research, retrieved 15 March 2011</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wolfram+Alpha&rft.atitle=Calculation+of+Integrate%28ln%28x%29%29&rft_id=http%3A%2F%2Fwww.wolframalpha.com%2Finput%2F%3Fi%3DIntegrate%28ln%28x%29%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> Abramowitz & Stegun, eds. <a href="#CITEREFAbramowitzStegun1972">1972</a>, p. 69 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourant1988" class="citation cs2">Courant, Richard (1988), Differential and integral calculus. Vol. I, Wiley Classics Library, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-60842-4" title="Special:BookSources/978-0-471-60842-4"><bdi>978-0-471-60842-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1009558">1009558</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+and+integral+calculus.+Vol.+I&rft.place=New+York&rft.series=Wiley+Classics+Library&rft.pub=John+Wiley+%26+Sons&rft.date=1988&rft.isbn=978-0-471-60842-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1009558%23id-name%3DMR&rft.aulast=Courant&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section III.6 </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHavil2003" class="citation cs2">Havil, Julian (2003), Gamma: Exploring Euler's Constant, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-09983-5" title="Special:BookSources/978-0-691-09983-5"><bdi>978-0-691-09983-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gamma%3A+Exploring+Euler%27s+Constant&rft.pub=Princeton+University+Press&rft.date=2003&rft.isbn=978-0-691-09983-5&rft.aulast=Havil&rft.aufirst=Julian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, sections 11.5 and 13.8 </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNomizu1996" class="citation cs2"><a href="/wiki/Katsumi_Nomizu" title="Katsumi Nomizu">Nomizu, Katsumi</a> (1996), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uDDxdu0lrWAC&pg=PA21">Selected papers on number theory and algebraic geometry</a>, vol. 172, Providence, RI: AMS Bookstore, p. 21, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0445-2" title="Special:BookSources/978-0-8218-0445-2"><bdi>978-0-8218-0445-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Selected+papers+on+number+theory+and+algebraic+geometry&rft.place=Providence%2C+RI&rft.pages=21&rft.pub=AMS+Bookstore&rft.date=1996&rft.isbn=978-0-8218-0445-2&rft.aulast=Nomizu&rft.aufirst=Katsumi&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuDDxdu0lrWAC%26pg%3DPA21&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaker1975" class="citation cs2"><a href="/wiki/Alan_Baker_(mathematician)" title="Alan Baker (mathematician)">Baker, Alan</a> (1975), Transcendental number theory, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-20461-3" title="Special:BookSources/978-0-521-20461-3"><bdi>978-0-521-20461-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transcendental+number+theory&rft.pub=Cambridge+University+Press&rft.date=1975&rft.isbn=978-0-521-20461-3&rft.aulast=Baker&rft.aufirst=Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, p. 10 </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMuller2006" class="citation cs2">Muller, Jean-Michel (2006), Elementary functions (2nd ed.), Boston, MA: Birkhäuser Boston, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4372-0" title="Special:BookSources/978-0-8176-4372-0"><bdi>978-0-8176-4372-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+functions&rft.place=Boston%2C+MA&rft.edition=2nd&rft.pub=Birkh%C3%A4user+Boston&rft.date=2006&rft.isbn=978-0-8176-4372-0&rft.aulast=Muller&rft.aufirst=Jean-Michel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, sections 4.2.2 (p. 72) and 5.5.2 (p. 95) </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartCheneyLawson1968" class="citation cs2">Hart; Cheney; Lawson; et al. (1968), Computer Approximations, SIAM Series in Applied Mathematics, New York: John Wiley</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Approximations&rft.place=New+York&rft.series=SIAM+Series+in+Applied+Mathematics&rft.pub=John+Wiley&rft.date=1968&rft.au=Hart&rft.au=Cheney&rft.au=Lawson&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 6.3, pp. 105–11 </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhangDelgado-FriasVassiliadis1994" class="citation cs2">Zhang, M.; Delgado-Frias, J.G.; Vassiliadis, S. (1994), "Table driven Newton scheme for high precision logarithm generation", IEE Proceedings - Computers and Digital Techniques, 141 (5): 281–92, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1049%2Fip-cdt%3A19941268">10.1049/ip-cdt:19941268</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1350-2387">1350-2387</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEE+Proceedings+-+Computers+and+Digital+Techniques&rft.atitle=Table+driven+Newton+scheme+for+high+precision+logarithm+generation&rft.volume=141&rft.issue=5&rft.pages=281-92&rft.date=1994&rft_id=info%3Adoi%2F10.1049%2Fip-cdt%3A19941268&rft.issn=1350-2387&rft.aulast=Zhang&rft.aufirst=M.&rft.au=Delgado-Frias%2C+J.G.&rft.au=Vassiliadis%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 1 for an overview </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeggitt1962" class="citation cs2">Meggitt, J.E. (April 1962), "Pseudo Division and Pseudo Multiplication Processes", IBM Journal of Research and Development, 6 (2): 210–26, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1147%2Frd.62.0210">10.1147/rd.62.0210</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:19387286">19387286</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IBM+Journal+of+Research+and+Development&rft.atitle=Pseudo+Division+and+Pseudo+Multiplication+Processes&rft.volume=6&rft.issue=2&rft.pages=210-26&rft.date=1962-04&rft_id=info%3Adoi%2F10.1147%2Frd.62.0210&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A19387286%23id-name%3DS2CID&rft.aulast=Meggitt&rft.aufirst=J.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKahan2001" class="citation cs2"><a href="/wiki/William_Kahan" title="William Kahan">Kahan, W.</a> (20 May 2001), Pseudo-Division Algorithms for Floating-Point Logarithms and Exponentials</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pseudo-Division+Algorithms+for+Floating-Point+Logarithms+and+Exponentials&rft.date=2001-05-20&rft.aulast=Kahan&rft.aufirst=W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-AbramowitzStegunp.68-54">^ <a href="#cite_ref-AbramowitzStegunp.68_54-0">a</a> <a href="#cite_ref-AbramowitzStegunp.68_54-1">b</a> Abramowitz & Stegun, eds. <a href="#CITEREFAbramowitzStegun1972">1972</a>, p. 68 </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSasakiKanada1982" class="citation cs2">Sasaki, T.; Kanada, Y. (1982), <a rel="nofollow" class="external text" href="http://ci.nii.ac.jp/naid/110002673332">"Practically fast multiple-precision evaluation of log(x)"</a>, Journal of Information Processing, 5 (4): 247–50, retrieved 30 March 2011</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Information+Processing&rft.atitle=Practically+fast+multiple-precision+evaluation+of+log%28x%29&rft.volume=5&rft.issue=4&rft.pages=247-50&rft.date=1982&rft.aulast=Sasaki&rft.aufirst=T.&rft.au=Kanada%2C+Y.&rft_id=http%3A%2F%2Fci.nii.ac.jp%2Fnaid%2F110002673332&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhrendt1999" class="citation cs2">Ahrendt, Timm (1999), "Fast Computations of the Exponential Function", Stacs 99, Lecture notes in computer science, vol. 1564, Berlin, New York: Springer, pp. 302–12, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-49116-3_28">10.1007/3-540-49116-3_28</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-65691-3" title="Special:BookSources/978-3-540-65691-3"><bdi>978-3-540-65691-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fast+Computations+of+the+Exponential+Function&rft.btitle=Stacs+99&rft.place=Berlin%2C+New+York&rft.series=Lecture+notes+in+computer+science&rft.pages=302-12&rft.pub=Springer&rft.date=1999&rft_id=info%3Adoi%2F10.1007%2F3-540-49116-3_28&rft.isbn=978-3-540-65691-3&rft.aulast=Ahrendt&rft.aufirst=Timm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHillis1989" class="citation cs2"><a href="/wiki/Danny_Hillis" title="Danny Hillis">Hillis, Danny</a> (15 January 1989), "Richard Feynman and The Connection Machine", Physics Today, 42 (2): 78, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989PhT....42b..78H">1989PhT....42b..78H</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.881196">10.1063/1.881196</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Richard+Feynman+and+The+Connection+Machine&rft.volume=42&rft.issue=2&rft.pages=78&rft.date=1989-01-15&rft_id=info%3Adoi%2F10.1063%2F1.881196&rft_id=info%3Abibcode%2F1989PhT....42b..78H&rft.aulast=Hillis&rft.aufirst=Danny&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> Maor <a href="#CITEREFMaor2009">2009</a>, p. 135 </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrey2006" class="citation cs2">Frey, Bruce (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HOPyiNb9UqwC&pg=PA275">Statistics hacks</a>, Hacks Series, Sebastopol, CA: <a href="/wiki/O%27Reilly_Media" title="O'Reilly Media">O'Reilly</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-596-10164-0" title="Special:BookSources/978-0-596-10164-0"><bdi>978-0-596-10164-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Statistics+hacks&rft.place=Sebastopol%2C+CA&rft.series=Hacks+Series&rft.pub=O%27Reilly&rft.date=2006&rft.isbn=978-0-596-10164-0&rft.aulast=Frey&rft.aufirst=Bruce&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DHOPyiNb9UqwC%26pg%3DPA275&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, chapter 6, section 64 </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRicciardi1990" class="citation cs2">Ricciardi, Luigi M. (1990), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Cw4NAQAAIAAJ">Lectures in applied mathematics and informatics</a>, Manchester: Manchester University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7190-2671-3" title="Special:BookSources/978-0-7190-2671-3"><bdi>978-0-7190-2671-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+in+applied+mathematics+and+informatics&rft.place=Manchester&rft.pub=Manchester+University+Press&rft.date=1990&rft.isbn=978-0-7190-2671-3&rft.aulast=Ricciardi&rft.aufirst=Luigi+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCw4NAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, p. 21, section 1.3.2 </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSankaran2001" class="citation book cs2">Sankaran, C. (2001), "7.5.1 Decibel (dB)", Power Quality, Taylor & Francis, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780849310409" title="Special:BookSources/9780849310409"><bdi>9780849310409</bdi></a>, <q>The decibel is used to express the ratio between two quantities. The quantities may be voltage, current, or power.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=7.5.1+Decibel+%28dB%29&rft.btitle=Power+Quality&rft.pub=Taylor+%26+Francis&rft.date=2001&rft.isbn=9780849310409&rft.aulast=Sankaran&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaling2007" class="citation cs2">Maling, George C. (2007), "Noise", in Rossing, Thomas D. (ed.), Springer handbook of acoustics, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-30446-5" title="Special:BookSources/978-0-387-30446-5"><bdi>978-0-387-30446-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Noise&rft.btitle=Springer+handbook+of+acoustics&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=2007&rft.isbn=978-0-387-30446-5&rft.aulast=Maling&rft.aufirst=George+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 23.0.2 </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTashev2009" class="citation cs2">Tashev, Ivan Jelev (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=plll9smnbOIC&pg=PA48">Sound Capture and Processing: Practical Approaches</a>, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, p. 98, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-31983-3" title="Special:BookSources/978-0-470-31983-3"><bdi>978-0-470-31983-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sound+Capture+and+Processing%3A+Practical+Approaches&rft.place=New+York&rft.pages=98&rft.pub=John+Wiley+%26+Sons&rft.date=2009&rft.isbn=978-0-470-31983-3&rft.aulast=Tashev&rft.aufirst=Ivan+Jelev&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dplll9smnbOIC%26pg%3DPA48&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChui1997" class="citation cs2">Chui, C.K. (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=N06Gu433PawC&pg=PA180">Wavelets: a mathematical tool for signal processing</a>, SIAM monographs on mathematical modeling and computation, Philadelphia: <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-384-8" title="Special:BookSources/978-0-89871-384-8"><bdi>978-0-89871-384-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wavelets%3A+a+mathematical+tool+for+signal+processing&rft.place=Philadelphia&rft.series=SIAM+monographs+on+mathematical+modeling+and+computation&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1997&rft.isbn=978-0-89871-384-8&rft.aulast=Chui&rft.aufirst=C.K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DN06Gu433PawC%26pg%3DPA180&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrauderEvansNoell2008" class="citation cs2">Crauder, Bruce; Evans, Benny; Noell, Alan (2008), Functions and Change: A Modeling Approach to College Algebra (4th ed.), Boston: Cengage Learning, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-547-15669-9" title="Special:BookSources/978-0-547-15669-9"><bdi>978-0-547-15669-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functions+and+Change%3A+A+Modeling+Approach+to+College+Algebra&rft.place=Boston&rft.edition=4th&rft.pub=Cengage+Learning&rft.date=2008&rft.isbn=978-0-547-15669-9&rft.aulast=Crauder&rft.aufirst=Bruce&rft.au=Evans%2C+Benny&rft.au=Noell%2C+Alan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 4.4. </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBradt2004" class="citation cs2">Bradt, Hale (2004), Astronomy methods: a physical approach to astronomical observations, Cambridge Planetary Science, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-53551-9" title="Special:BookSources/978-0-521-53551-9"><bdi>978-0-521-53551-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Astronomy+methods%3A+a+physical+approach+to+astronomical+observations&rft.series=Cambridge+Planetary+Science&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-0-521-53551-9&rft.aulast=Bradt&rft.aufirst=Hale&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 8.3, p. 231 </li> <li id="cite_note-Jens-68"><a href="#cite_ref-Jens_68-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNørby,_Jens2000" class="citation journal cs2">Nørby, Jens (2000), "The origin and the meaning of the little p in pH", Trends in Biochemical Sciences, 25 (1): 36–37, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0968-0004%2899%2901517-0">10.1016/S0968-0004(99)01517-0</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10637613">10637613</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trends+in+Biochemical+Sciences&rft.atitle=The+origin+and+the+meaning+of+the+little+p+in+pH&rft.volume=25&rft.issue=1&rft.pages=36-37&rft.date=2000&rft_id=info%3Adoi%2F10.1016%2FS0968-0004%2899%2901517-0&rft_id=info%3Apmid%2F10637613&rft.au=N%C3%B8rby%2C+Jens&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIUPAC1997" class="citation cs2"><a href="/wiki/IUPAC" class="mw-redirect" title="IUPAC">IUPAC</a> (1997), A. 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Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures", Science, 320 (5880): 1217–20, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008Sci...320.1217D">2008Sci...320.1217D</a>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.362.2390">10.1.1.362.2390</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1156540">10.1126/science.1156540</a>, <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2610411">2610411</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18511690">18511690</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Log+or+Linear%3F+Distinct+Intuitions+of+the+Number+Scale+in+Western+and+Amazonian+Indigene+Cultures&rft.volume=320&rft.issue=5880&rft.pages=1217-20&rft.date=2008&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2610411%23id-name%3DPMC&rft_id=info%3Abibcode%2F2008Sci...320.1217D&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.362.2390%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F18511690&rft_id=info%3Adoi%2F10.1126%2Fscience.1156540&rft.aulast=Dehaene&rft.aufirst=Stanislas&rft.au=Izard%2C+V%C3%A9ronique&rft.au=Spelke%2C+Elizabeth&rft.au=Pica%2C+Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBreiman1992" class="citation cs2">Breiman, Leo (1992), Probability, Classics in applied mathematics, Philadelphia: <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">Society for Industrial and Applied Mathematics</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-296-4" title="Special:BookSources/978-0-89871-296-4"><bdi>978-0-89871-296-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability&rft.place=Philadelphia&rft.series=Classics+in+applied+mathematics&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1992&rft.isbn=978-0-89871-296-4&rft.aulast=Breiman&rft.aufirst=Leo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 12.9 </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAitchisonBrown1969" class="citation cs2">Aitchison, J.; Brown, J.A.C. 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Schopfer, Peter (1995), <a rel="nofollow" class="external text" href="https://archive.org/details/plantphysiology0000mohr">Plant physiology</a>, Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-58016-4" title="Special:BookSources/978-3-540-58016-4"><bdi>978-3-540-58016-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plant+physiology&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=978-3-540-58016-4&rft.aulast=Mohr&rft.aufirst=Hans&rft.au=Schopfer%2C+Peter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fplantphysiology0000mohr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, chapter 19, p. 298 </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEco1989" class="citation cs2"><a href="/wiki/Umberto_Eco" title="Umberto Eco">Eco, Umberto</a> (1989), The open work, <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-63976-8" title="Special:BookSources/978-0-674-63976-8"><bdi>978-0-674-63976-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+open+work&rft.pub=Harvard+University+Press&rft.date=1989&rft.isbn=978-0-674-63976-8&rft.aulast=Eco&rft.aufirst=Umberto&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section III.I </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSprott2010" class="citation cs2">Sprott, Julien Clinton (2010), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=buILBDre9S4C">"Elegant Chaos: Algebraically Simple Chaotic Flows"</a>, Elegant Chaos: Algebraically Simple Chaotic Flows. Edited by Sprott Julien Clinton. Published by World Scientific Publishing Co. Pte. Ltd, New Jersey: <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010ecas.book.....S">2010ecas.book.....S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F7183">10.1142/7183</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-283-881-0" title="Special:BookSources/978-981-283-881-0"><bdi>978-981-283-881-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Elegant+Chaos%3A+Algebraically+Simple+Chaotic+Flows.+Edited+by+Sprott+Julien+Clinton.+Published+by+World+Scientific+Publishing+Co.+Pte.+Ltd&rft.atitle=Elegant+Chaos%3A+Algebraically+Simple+Chaotic+Flows&rft.date=2010&rft_id=info%3Adoi%2F10.1142%2F7183&rft_id=info%3Abibcode%2F2010ecas.book.....S&rft.isbn=978-981-283-881-0&rft.aulast=Sprott&rft.aufirst=Julien+Clinton&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbuILBDre9S4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 1.9 </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelmberg2007" class="citation cs2">Helmberg, Gilbert (2007), Getting acquainted with fractals, De Gruyter Textbook, Berlin, New York: Walter de Gruyter, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-019092-2" title="Special:BookSources/978-3-11-019092-2"><bdi>978-3-11-019092-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Getting+acquainted+with+fractals&rft.place=Berlin%2C+New+York&rft.series=De+Gruyter+Textbook&rft.pub=Walter+de+Gruyter&rft.date=2007&rft.isbn=978-3-11-019092-2&rft.aulast=Helmberg&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWright2009" class="citation cs2">Wright, David (2009), Mathematics and music, Providence, RI: AMS Bookstore, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4873-9" title="Special:BookSources/978-0-8218-4873-9"><bdi>978-0-8218-4873-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+music&rft.place=Providence%2C+RI&rft.pub=AMS+Bookstore&rft.date=2009&rft.isbn=978-0-8218-4873-9&rft.aulast=Wright&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, chapter 5 </li> <li id="cite_note-95"><a href="#cite_ref-95">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatemanDiamond2004" class="citation cs2">Bateman, P.T.; Diamond, Harold G. (2004), Analytic number theory: an introductory course, New Jersey: <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-256-080-3" title="Special:BookSources/978-981-256-080-3"><bdi>978-981-256-080-3</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/492669517">492669517</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analytic+number+theory%3A+an+introductory+course&rft.place=New+Jersey&rft.pub=World+Scientific&rft.date=2004&rft_id=info%3Aoclcnum%2F492669517&rft.isbn=978-981-256-080-3&rft.aulast=Bateman&rft.aufirst=P.T.&rft.au=Diamond%2C+Harold+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, theorem 4.1 </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> P. T. Bateman & Diamond <a href="#CITEREFBatemanDiamond2004">2004</a>, Theorem 8.15 </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSlomson1991" class="citation cs2">Slomson, Alan B. (1991), An introduction to combinatorics, London: <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-35370-3" title="Special:BookSources/978-0-412-35370-3"><bdi>978-0-412-35370-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+combinatorics&rft.place=London&rft.pub=CRC+Press&rft.date=1991&rft.isbn=978-0-412-35370-3&rft.aulast=Slomson&rft.aufirst=Alan+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, chapter 4 </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGanguly2005" class="citation cs2">Ganguly, S. (2005), Elements of Complex Analysis, Kolkata: Academic Publishers, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-87504-86-3" title="Special:BookSources/978-81-87504-86-3"><bdi>978-81-87504-86-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Complex+Analysis&rft.place=Kolkata&rft.pub=Academic+Publishers&rft.date=2005&rft.isbn=978-81-87504-86-3&rft.aulast=Ganguly&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, Definition 1.6.3 </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNevanlinnaPaatero2007" class="citation cs2"><a href="/wiki/Rolf_Nevanlinna" title="Rolf Nevanlinna">Nevanlinna, Rolf Herman</a>; Paatero, Veikko (2007), "Introduction to complex analysis", London: Hilger, Providence, RI: AMS Bookstore, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974aitc.book.....W">1974aitc.book.....W</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4399-4" title="Special:BookSources/978-0-8218-4399-4"><bdi>978-0-8218-4399-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=London%3A+Hilger&rft.atitle=Introduction+to+complex+analysis&rft.date=2007&rft_id=info%3Abibcode%2F1974aitc.book.....W&rft.isbn=978-0-8218-4399-4&rft.aulast=Nevanlinna&rft.aufirst=Rolf+Herman&rft.au=Paatero%2C+Veikko&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 5.9 </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMooreHadlock1991" class="citation cs2">Moore, Theral Orvis; Hadlock, Edwin H. (1991), Complex analysis, Singapore: <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-0246-0" title="Special:BookSources/978-981-02-0246-0"><bdi>978-981-02-0246-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complex+analysis&rft.place=Singapore&rft.pub=World+Scientific&rft.date=1991&rft.isbn=978-981-02-0246-0&rft.aulast=Moore&rft.aufirst=Theral+Orvis&rft.au=Hadlock%2C+Edwin+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 1.2 </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilde2006" class="citation cs2">Wilde, Ivan Francis (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vrWES2W6vG0C&q=complex+logarithm&pg=PA97">Lecture notes on complex analysis</a>, London: Imperial College Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-86094-642-4" title="Special:BookSources/978-1-86094-642-4"><bdi>978-1-86094-642-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lecture+notes+on+complex+analysis&rft.place=London&rft.pub=Imperial+College+Press&rft.date=2006&rft.isbn=978-1-86094-642-4&rft.aulast=Wilde&rft.aufirst=Ivan+Francis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DvrWES2W6vG0C%26q%3Dcomplex%2Blogarithm%26pg%3DPA97&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, theorem 6.1. </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHigham2008" class="citation cs2"><a href="/wiki/Nicholas_Higham" title="Nicholas Higham">Higham, Nicholas</a> (2008), Functions of Matrices. Theory and Computation, Philadelphia, PA: <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-646-7" title="Special:BookSources/978-0-89871-646-7"><bdi>978-0-89871-646-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functions+of+Matrices.+Theory+and+Computation&rft.place=Philadelphia%2C+PA&rft.pub=SIAM&rft.date=2008&rft.isbn=978-0-89871-646-7&rft.aulast=Higham&rft.aufirst=Nicholas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, chapter 11. </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeukirch1999" class="citation book cs2"><a href="/wiki/J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Neukirch, Jürgen</a> (1999), Algebraische Zahlentheorie, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-65399-8" title="Special:BookSources/978-3-540-65399-8"><bdi>978-3-540-65399-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859">1697859</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0956.11021">0956.11021</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraische+Zahlentheorie&rft.place=Berlin&rft.series=%3Cspan+title%3D%22German-language+text%22%3E%3Ci+lang%3D%22de%22%3EGrundlehren+der+mathematischen+Wissenschaften%3C%2Fi%3E%3C%2Fspan%3ECategory%3AArticles+containing+German-language+text&rft.pub=Springer-Verlag&rft.date=1999&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0956.11021%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1697859%23id-name%3DMR&rft.isbn=978-3-540-65399-8&rft.aulast=Neukirch&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section II.5. </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHancockMartinSabin2009" class="citation cs2">Hancock, Edwin R.; Martin, Ralph R.; Sabin, Malcolm A. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0cqCy9x7V_QC&pg=PA379">Mathematics of Surfaces XIII: 13th IMA International Conference York, UK, September 7–9, 2009 Proceedings</a>, Springer, p. 379, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-03595-1" title="Special:BookSources/978-3-642-03595-1"><bdi>978-3-642-03595-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+of+Surfaces+XIII%3A+13th+IMA+International+Conference+York%2C+UK%2C+September+7%E2%80%939%2C+2009+Proceedings&rft.pages=379&rft.pub=Springer&rft.date=2009&rft.isbn=978-3-642-03595-1&rft.aulast=Hancock&rft.aufirst=Edwin+R.&rft.au=Martin%2C+Ralph+R.&rft.au=Sabin%2C+Malcolm+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0cqCy9x7V_QC%26pg%3DPA379&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-106"><a href="#cite_ref-106">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStinson2006" class="citation cs2">Stinson, Douglas Robert (2006), Cryptography: Theory and Practice (3rd ed.), London: <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-508-5" title="Special:BookSources/978-1-58488-508-5"><bdi>978-1-58488-508-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cryptography%3A+Theory+and+Practice&rft.place=London&rft.edition=3rd&rft.pub=CRC+Press&rft.date=2006&rft.isbn=978-1-58488-508-5&rft.aulast=Stinson&rft.aufirst=Douglas+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLidlNiederreiter1997" class="citation cs2">Lidl, Rudolf; <a href="/wiki/Harald_Niederreiter" title="Harald Niederreiter">Niederreiter, Harald</a> (1997), <a rel="nofollow" class="external text" href="https://archive.org/details/finitefields0000lidl_a8r3">Finite fields</a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-39231-0" title="Special:BookSources/978-0-521-39231-0"><bdi>978-0-521-39231-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+fields&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=978-0-521-39231-0&rft.aulast=Lidl&rft.aufirst=Rudolf&rft.au=Niederreiter%2C+Harald&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffinitefields0000lidl_a8r3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"> </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCorlessGonnetHareJeffrey1996" class="citation cs2">Corless, R.; 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Cherkassky, Vladimir S.; Mulier, Filip (2007), Learning from data: concepts, theory, and methods, Wiley series on adaptive and learning systems for signal processing, communications, and control, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-68182-3" title="Special:BookSources/978-0-471-68182-3"><bdi>978-0-471-68182-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Learning+from+data%3A+concepts%2C+theory%2C+and+methods&rft.place=New+York&rft.series=Wiley+series+on+adaptive+and+learning+systems+for+signal+processing%2C+communications%2C+and+control&rft.pub=John+Wiley+%26+Sons&rft.date=2007&rft.isbn=978-0-471-68182-3&rft.aulast=Cherkassky&rft.aufirst=Vladimir&rft.au=Cherkassky%2C+Vladimir+S.&rft.au=Mulier%2C+Filip&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, p. 357 </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1998" class="citation cs2"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1998), General topology. Chapters 5–10, Elements of Mathematics, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-64563-4" title="Special:BookSources/978-3-540-64563-4"><bdi>978-3-540-64563-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1726872">1726872</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+topology.+Chapters+5%E2%80%9310&rft.place=Berlin%2C+New+York&rft.series=Elements+of+Mathematics&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=978-3-540-64563-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1726872%23id-name%3DMR&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section V.4.1 </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmbartzumian1990" class="citation cs2"><a href="/wiki/Rouben_V._Ambartzumian" title="Rouben V. Ambartzumian">Ambartzumian, R.V.</a> (1990), <a rel="nofollow" class="external text" href="https://archive.org/details/factorizationcal0000amba">Factorization calculus and geometric probability</a>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-34535-4" title="Special:BookSources/978-0-521-34535-4"><bdi>978-0-521-34535-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Factorization+calculus+and+geometric+probability&rft.pub=Cambridge+University+Press&rft.date=1990&rft.isbn=978-0-521-34535-4&rft.aulast=Ambartzumian&rft.aufirst=R.V.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffactorizationcal0000amba&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 1.4 </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEsnaultViehweg1992" class="citation cs2">Esnault, Hélène; Viehweg, Eckart (1992), Lectures on vanishing theorems, DMV Seminar, vol. 20, Basel, Boston: Birkhäuser Verlag, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.178.3227">10.1.1.178.3227</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-0348-8600-0">10.1007/978-3-0348-8600-0</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-2822-1" title="Special:BookSources/978-3-7643-2822-1"><bdi>978-3-7643-2822-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1193913">1193913</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+vanishing+theorems&rft.place=Basel%2C+Boston&rft.series=DMV+Seminar&rft.pub=Birkh%C3%A4user+Verlag&rft.date=1992&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.178.3227%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1193913%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-0348-8600-0&rft.isbn=978-3-7643-2822-1&rft.aulast=Esnault&rft.aufirst=H%C3%A9l%C3%A8ne&rft.au=Viehweg%2C+Eckart&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">, section 2 </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFApostol2010" class="citation cs2">Apostol, T.M. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/25.12">"Logarithm"</a>, in <a href="/wiki/Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="/wiki/Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19225-5" title="Special:BookSources/978-0-521-19225-5"><bdi>978-0-521-19225-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Logarithm&rft.btitle=NIST+Handbook+of+Mathematical+Functions&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-19225-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2723248%23id-name%3DMR&rft.aulast=Apostol&rft.aufirst=T.M.&rft_id=http%3A%2F%2Fdlmf.nist.gov%2F25.12&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988">. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Logarithm&action=edit&section=42" title="Edit section: External links">edit</a>]</div> <ul><li><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Logarithm" class="extiw" title="commons:Category:Logarithm">Logarithm</a> at Wikimedia Commons</li> <li><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/16px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/24px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/32px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a> The dictionary definition of <a href="https://en.wiktionary.org/wiki/Special:Search/logarithm" class="extiw" title="wiktionary:Special:Search/logarithm">logarithm</a> at Wiktionary</li> <li><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/13px-Wikiquote-logo.svg.png" decoding="async" width="13" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/20px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/27px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a> Quotations related to <a href="https://en.wikiquote.org/wiki/History_of_logarithms" class="extiw" title="wikiquote:History of logarithms">History of logarithms</a> at Wikiquote</li> <li><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a> <a href="https://en.wikiversity.org/wiki/Speak_Math_Now!/Week_9:_Six_rules_of_Exponents/Logarithms" class="extiw" title="v:Speak Math Now!/Week 9: Six rules of Exponents/Logarithms">A lesson on logarithms can be found on Wikiversity</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs2"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Logarithm.html">"Logarithm"</a>, <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Logarithm&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLogarithm.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20121218200616/http://www.khanacademy.org/math/algebra/logarithms-tutorial">Khan Academy: Logarithms, free online micro lectures</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Logarithmic_function">"Logarithmic function"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Logarithmic+function&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DLogarithmic_function&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFColin_Byfleet" class="citation cs2">Colin Byfleet, <a rel="nofollow" class="external text" href="http://mediasite.oddl.fsu.edu/mediasite/Viewer/?peid=003298f9a02f468c8351c50488d6c479">Educational video on logarithms</a>, retrieved 12 October 2010</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Educational+video+on+logarithms&rft.au=Colin+Byfleet&rft_id=http%3A%2F%2Fmediasite.oddl.fsu.edu%2Fmediasite%2FViewer%2F%3Fpeid%3D003298f9a02f468c8351c50488d6c479&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdward_Wright" class="citation cs2">Edward Wright, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20021203005508/http://www.johnnapier.com/table_of_logarithms_001.htm">Translation of Napier's work on logarithms</a>, archived from <a rel="nofollow" class="external text" href="http://www.johnnapier.com/table_of_logarithms_001.htm">the original</a> on 3 December 2002, retrieved 12 October 2010</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Translation+of+Napier%27s+work+on+logarithms&rft.au=Edward+Wright&rft_id=http%3A%2F%2Fwww.johnnapier.com%2Ftable_of_logarithms_001.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogarithm" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGlaisher1911" class="citation encyclopaedia cs2">Glaisher, James Whitbread Lee (1911), <a class="external text" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Logarithm">"Logarithm" </a>, in <a href="/wiki/Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a> (ed.), <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a>, vol. 16 (11th ed.), Cambridge University Press, pp. 868–77</cite><span 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