Natural logarithm - Wikipedia

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id="toc-Inverse_of_exponential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Integral definition</span> </div> </a> <ul id="toc-Integral_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivative" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Derivative</span> </div> </a> <ul id="toc-Derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Series" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Series"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Series</span> </div> </a> <ul id="toc-Series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_natural_logarithm_in_integration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_natural_logarithm_in_integration"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>The natural logarithm in integration</span> </div> </a> <ul id="toc-The_natural_logarithm_in_integration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Efficient_computation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Efficient_computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Efficient computation</span> </div> </a> <button aria-controls="toc-Efficient_computation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Efficient computation subsection</span> </button> <ul id="toc-Efficient_computation-sublist" class="vector-toc-list"> <li id="toc-Natural_logarithm_of_10" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Natural_logarithm_of_10"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Natural logarithm of 10</span> </div> </a> <ul id="toc-Natural_logarithm_of_10-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-High_precision" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#High_precision"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>High precision</span> </div> </a> <ul id="toc-High_precision-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"> <span class="vector-toc-numb">8.3</span> <span>Computational complexity</span> </div> </a> <ul id="toc-Computational_complexity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Continued_fractions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Continued_fractions"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Continued fractions</span> </div> </a> <ul id="toc-Continued_fractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_logarithms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Complex_logarithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Complex logarithms</span> </div> </a> <ul id="toc-Complex_logarithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Notes</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-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" title="Table of Contents" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Natural logarithm</span></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 62 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-62" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">62 languages</span> </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/Natuurlike_logaritme" title="Natuurlike logaritme – Afrikaans" lang="af" hreflang="af" data-title="Natuurlike logaritme" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></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_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A" title="لوغاريتم طبيعي – Arabic" lang="ar" hreflang="ar" data-title="لوغاريتم طبيعي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Natural_loqarifm" title="Natural loqarifm – Azerbaijani" lang="az" hreflang="az" data-title="Natural loqarifm" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C5%AB-ji%C3%A2n_t%C3%B9i-s%C3%B2%CD%98" title="Chū-jiân tùi-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Chū-jiân tùi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D0%BB%D0%B0%D0%B3%D0%B0%D1%80%D1%8B%D1%84%D0%BC" title="Натуральны лагарыфм – Belarusian" lang="be" hreflang="be" data-title="Натуральны лагарыфм" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D0%B5%D0%BD_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D1%8A%D0%BC" title="Натурален логаритъм – Bulgarian" lang="bg" hreflang="bg" data-title="Натурален логаритъм" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prirodni_logaritam" title="Prirodni logaritam – Bosnian" lang="bs" hreflang="bs" data-title="Prirodni logaritam" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Logaritm_neperian" title="Logaritm neperian – Breton" lang="br" hreflang="br" data-title="Logaritm neperian" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Logaritme_natural" title="Logaritme natural – Catalan" lang="ca" hreflang="ca" data-title="Logaritme natural" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D0%BB%C4%83_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Натураллă логарифм – Chuvash" lang="cv" hreflang="cv" data-title="Натураллă логарифм" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Naturlig_logaritme" title="Naturlig logaritme – Danish" lang="da" hreflang="da" data-title="Naturlig logaritme" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Logarithmus#Natürlicher_Logarithmus" title="Logarithmus – German" lang="de" hreflang="de" data-title="Logarithmus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Naturaallogaritm" title="Naturaallogaritm – Estonian" lang="et" hreflang="et" data-title="Naturaallogaritm" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A6%CF%85%CF%83%CE%B9%CE%BA%CF%8C%CF%82_%CE%BB%CE%BF%CE%B3%CE%AC%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82" title="Φυσικός λογάριθμος – Greek" lang="el" hreflang="el" data-title="Φυσικός λογάριθμος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Logaritmo_natural" title="Logaritmo natural – Spanish" lang="es" hreflang="es" data-title="Logaritmo natural" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Logaritmo_natural" title="Logaritmo natural – Basque" lang="eu" hreflang="eu" data-title="Logaritmo natural" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></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_%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C" title="لگاریتم طبیعی – Persian" lang="fa" hreflang="fa" data-title="لگاریتم طبیعی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/N%C3%A1tt%C3%BArlig_logaritma" title="Náttúrlig logaritma – Faroese" lang="fo" hreflang="fo" data-title="Náttúrlig logaritma" data-language-autonym="Føroyskt" data-language-local-name="Faroese" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Logarithme_n%C3%A9p%C3%A9rien" title="Logarithme népérien – French" lang="fr" hreflang="fr" data-title="Logarithme népérien" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Logaritmo_natural" title="Logaritmo natural – Galician" lang="gl" hreflang="gl" data-title="Logaritmo natural" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%97%B0%EB%A1%9C%EA%B7%B8" title="자연로그 – Korean" lang="ko" hreflang="ko" data-title="자연로그" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6_%D5%AC%D5%B8%D5%A3%D5%A1%D6%80%D5%AB%D5%A9%D5%B4" title="Բնական լոգարիթմ – Armenian" lang="hy" hreflang="hy" data-title="Բնական լոգարիթմ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%B2%E0%A4%98%E0%A5%81%E0%A4%97%E0%A4%A3%E0%A4%95" title="प्राकृतिक लघुगणक – Hindi" lang="hi" hreflang="hi" data-title="प्राकृतिक लघुगणक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logaritma_alami" title="Logaritma alami – Indonesian" lang="id" hreflang="id" data-title="Logaritma alami" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logaritmo_naturale" title="Logaritmo naturale – Italian" lang="it" hreflang="it" data-title="Logaritmo naturale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></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_%D7%98%D7%91%D7%A2%D7%99" title="לוגריתם טבעי – Hebrew" lang="he" hreflang="he" data-title="לוגריתם טבעי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%90%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%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="ნატურალური ლოგარითმი – Georgian" lang="ka" hreflang="ka" data-title="ნატურალური ლოგარითმი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Натурал логарифм – Kazakh" lang="kk" hreflang="kk" data-title="Натурал логарифм" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Natur%C4%81lais_logaritms" title="Naturālais logaritms – Latvian" lang="lv" hreflang="lv" data-title="Naturālais logaritms" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Nat%C5%ABrinis_logaritmas" title="Natūrinis logaritmas – Lithuanian" lang="lt" hreflang="lt" data-title="Natūrinis logaritmas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Logaritm_natural" title="Logaritm natural – Lombard" lang="lmo" hreflang="lmo" data-title="Logaritm natural" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Term%C3%A9szetes_logaritmus" title="Természetes logaritmus – Hungarian" lang="hu" hreflang="hu" data-title="Természetes logaritmus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%B5%D0%BD_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Природен логаритам – Macedonian" lang="mk" hreflang="mk" data-title="Природен логаритам" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logaritma_asli" title="Logaritma asli – Malay" lang="ms" hreflang="ms" data-title="Logaritma asli" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Natuurlijke_logaritme" title="Natuurlijke logaritme – Dutch" lang="nl" hreflang="nl" data-title="Natuurlijke logaritme" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%AF%BE%E6%95%B0" title="自然対数 – Japanese" lang="ja" hreflang="ja" data-title="自然対数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Naturlig_logaritme" title="Naturlig logaritme – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Naturlig logaritme" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Logaritme_neperian" title="Logaritme neperian – Occitan" lang="oc" hreflang="oc" data-title="Logaritme neperian" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Natural_logarifm" title="Natural logarifm – Uzbek" lang="uz" hreflang="uz" data-title="Natural logarifm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Logaritm_%C3%ABd_Napier" title="Logaritm ëd Napier – Piedmontese" lang="pms" hreflang="pms" data-title="Logaritm ëd Napier" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logarytm_naturalny" title="Logarytm naturalny – Polish" lang="pl" hreflang="pl" data-title="Logarytm naturalny" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Logaritmo_natural" title="Logaritmo natural – Portuguese" lang="pt" hreflang="pt" data-title="Logaritmo natural" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Logaritm_natural" title="Logaritm natural – Romanian" lang="ro" hreflang="ro" data-title="Logaritm natural" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Натуральный логарифм – Russian" lang="ru" hreflang="ru" data-title="Натуральный логарифм" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logaritmi_natyror" title="Logaritmi natyror – Albanian" lang="sq" hreflang="sq" data-title="Logaritmi natyror" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%9A%E0%B7%98%E0%B6%AD%E0%B7%92_%E0%B6%BD%E0%B6%9D%E0%B7%94%E0%B6%9C%E0%B6%AB%E0%B6%9A" title="ප්‍රකෘති ලඝුගණක – Sinhala" lang="si" hreflang="si" data-title="ප්‍රකෘති ලඝුගණක" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Naravni_logaritem" title="Naravni logaritem – Slovenian" lang="sl" hreflang="sl" data-title="Naravni logaritem" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%86%DA%AF%D8%A7%D8%B1%DB%8C%D8%AA%D9%85%DB%8C_%D8%B3%D8%B1%D9%88%D8%B4%D8%AA%DB%8C" title="لۆگاریتمی سروشتی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="لۆگاریتمی سروشتی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%BD%D0%B8_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Природни логаритам – Serbian" lang="sr" hreflang="sr" data-title="Природни логаритам" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Prirodni_logaritam" title="Prirodni logaritam – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Prirodni logaritam" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Luonnollinen_logaritmi" title="Luonnollinen logaritmi – Finnish" lang="fi" hreflang="fi" data-title="Luonnollinen logaritmi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Naturliga_logaritmen" title="Naturliga logaritmen – Swedish" lang="sv" hreflang="sv" data-title="Naturliga logaritmen" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%AE%E0%AE%9F%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%88" title="இயல் மடக்கை – Tamil" lang="ta" hreflang="ta" data-title="இயல் மடக்கை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Alugaritm_agaman" title="Alugaritm agaman – Tachelhit" lang="shi" hreflang="shi" data-title="Alugaritm agaman" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%AD%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%B4%E0%B8%97%E0%B8%B6%E0%B8%A1%E0%B8%98%E0%B8%A3%E0%B8%A3%E0%B8%A1%E0%B8%8A%E0%B8%B2%E0%B8%95%E0%B8%B4" title="ลอการิทึมธรรมชาติ – Thai" lang="th" hreflang="th" data-title="ลอการิทึมธรรมชาติ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Натуральний логарифм – Ukrainian" lang="uk" hreflang="uk" data-title="Натуральний логарифм" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%D8%AF%D8%B1%D8%AA%DB%8C_%D9%84%D8%A7%DA%AF%D8%B1%D8%AA%DA%BE%D9%85" title="قدرتی لاگرتھم – Urdu" lang="ur" hreflang="ur" data-title="قدرتی لاگرتھم" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Logarit_t%E1%BB%B1_nhi%C3%AAn" title="Logarit tự nhiên – Vietnamese" lang="vi" hreflang="vi" data-title="Logarit tự nhiên" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%B0%8D%E6%95%B8" title="自然對數 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="自然對數" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%AF%B9%E6%95%B0" title="自然对数 – Wu" lang="wuu" hreflang="wuu" data-title="自然对数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%B0%8D%E6%95%B8" title="自然對數 – Cantonese" lang="yue" hreflang="yue" data-title="自然對數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%B0%8D%E6%95%B8" title="自然對數 – Chinese" lang="zh" hreflang="zh" data-title="自然對數" data-language-autonym="中文" data-language-local-name="Chinese" 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<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">Logarithm to the base of the mathematical constant e</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">"Base e" redirects here. For the numbering system which uses "e" as its base, see <a href="/wiki/Non-integer_base_of_numeration#Base_e" title="Non-integer base of numeration">Non-integer base of numeration § Base e</a>.</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e0e0e0;padding:0.15em 0.5em 0.25em;font-weight:bold;">Natural logarithm</th></tr><tr><td colspan="2" class="infobox-image" style="padding-bottom:0.4em;"><span typeof="mw:File"><a href="/wiki/File:Log_(2).svg" class="mw-file-description"><img alt="Graph of part of the natural logarithm function." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Log_%282%29.svg/290px-Log_%282%29.svg.png" decoding="async" width="290" height="289" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Log_%282%29.svg/435px-Log_%282%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Log_%282%29.svg/580px-Log_%282%29.svg.png 2x" data-file-width="386" data-file-height="385" /></a></span><div class="infobox-caption">Graph of part of the natural logarithm function. The function slowly grows to positive infinity as <span class="texhtml mvar" style="font-style:italic;">x</span> increases, and slowly goes to negative infinity as <span class="texhtml mvar" style="font-style:italic;">x</span> approaches 0 ("slowly" as compared to any <a href="/wiki/Power_law" title="Power law">power law</a> of <span class="texhtml mvar" style="font-style:italic;">x</span>).</div></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">General information</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">General definition</th><td class="infobox-data"><span class="mwe-math-element" data-qid="Q204037"><a href="/w/index.php?title=Special:MathWikibase&qid=Q204037" style="color:inherit;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x=\log _{e}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x=\log _{e}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde508ef62045967e46c044124635b193e0f5054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.442ex; height:2.676ex;" alt="{\displaystyle \ln x=\log _{e}x}"></a></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Motivation of invention</th><td class="infobox-data"><a href="/wiki/Hyperbolic_logarithm" class="mw-redirect" title="Hyperbolic logarithm">hyperbola quadrature</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Fields of application</th><td class="infobox-data">Pure and applied mathematics</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Domain, codomain and image</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Domain_of_a_function" title="Domain of a function">Domain</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Codomain" title="Codomain">Codomain</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">Image</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Specific values</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Value at +∞</th><td class="infobox-data">+∞</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Value at <span class="texhtml mvar" style="font-style:italic;">e</span></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Value at 1</th><td class="infobox-data">0</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Value at 0</th><td class="infobox-data">-∞</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Specific features</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Asymptote" title="Asymptote">Asymptote</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Zero_of_a_function" title="Zero of a function">Root</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Inverse_function" title="Inverse function">Inverse</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb48cbff4a1363dbf36792710839f53bd5e02ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.269ex; height:2.009ex;" alt="{\displaystyle \exp x}"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Derivative" title="Derivative">Derivative</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {d}{dx}}\ln x={\dfrac {1}{x}},x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> <mo>,</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dfrac {d}{dx}}\ln x={\dfrac {1}{x}},x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/831d0950c285206114b05050189ddc9c570f9eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.314ex; height:5.509ex;" alt="{\displaystyle {\dfrac {d}{dx}}\ln x={\dfrac {1}{x}},x>0}"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \ln x\,dx=x\left(\ln x-1\right)+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \ln x\,dx=x\left(\ln x-1\right)+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ca1dd948a9f94c860ff1eaec58272f6a19df5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.06ex; height:5.676ex;" alt="{\displaystyle \int \ln x\,dx=x\left(\ln x-1\right)+C}"></span></td></tr></tbody></table> <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 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.sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1238256010">.mw-parser-output .e-mathematical-constant-sidebar{width:20em}.mw-parser-output .e-mathematical-constant-sidebar .sidebar-title-with-pretitle{font-size:130%}.mw-parser-output .e-mathematical-constant-sidebar .sidebar-heading{border-top:#aaa 1px solid}</style><table class="sidebar nomobile nowraplinks hlist e-mathematical-constant-sidebar"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:E_(mathematical_constant)" title="Category:E (mathematical constant)">a series of articles</a> on the</td></tr><tr><th class="sidebar-title-with-pretitle">mathematical constant <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></span></th></tr><tr><td class="sidebar-image"><span class="skin-invert-image" typeof="mw:File"><a href="/wiki/File:Euler%27s_formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/180px-Euler%27s_formula.svg.png" decoding="async" width="180" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/270px-Euler%27s_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/360px-Euler%27s_formula.svg.png 2x" data-file-width="760" data-file-height="782" /></a></span></td></tr><tr><th class="sidebar-heading"> Properties</th></tr><tr><td class="sidebar-content"> <ul><li><a class="mw-selflink selflink">Natural logarithm</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Applications</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Compound_interest" title="Compound interest">compound interest</a></li> <li><a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a></li> <li><a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Half-life" title="Half-life">half-lives</a> <ul><li>exponential <a href="/wiki/Exponential_growth" title="Exponential growth">growth</a> and <a href="/wiki/Exponential_decay" title="Exponential decay">decay</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Defining <span class="texhtml mvar" style="font-style:italic;">e</span></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Proof_that_e_is_irrational" title="Proof that e is irrational">proof that <span class="texhtml mvar" style="font-style:italic;">e</span> is irrational</a></li> <li><a href="/wiki/List_of_representations_of_e" title="List of representations of e">representations of <span class="texhtml mvar" style="font-style:italic;">e</span></a></li> <li><a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> People</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/John_Napier" title="John Napier">John Napier</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a><br /></li></ul></td> </tr><tr><th class="sidebar-heading"> Related topics</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Schanuel%27s_conjecture" title="Schanuel's conjecture">Schanuel's conjecture</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .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><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:E_(mathematical_constant)" title="Template:E (mathematical constant)"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:E_(mathematical_constant)" title="Template talk:E (mathematical constant)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:E_(mathematical_constant)" title="Special:EditPage/Template:E (mathematical constant)"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>natural logarithm</b> of a number is its <a href="/wiki/Logarithm" title="Logarithm">logarithm</a> to the <a href="/wiki/Base_of_a_logarithm" class="mw-redirect" title="Base of a logarithm">base</a> of the <a href="/wiki/Mathematical_constant" title="Mathematical constant">mathematical constant</a> <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)"><span class="texhtml mvar" style="font-style:italic;">e</span></a>, which is an <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> and <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a> number approximately equal to <span class="texhtml"><span class="nowrap"><span data-sort-value="7000271828182845899♠"></span>2.718<span style="margin-left:.25em;">281</span><span style="margin-left:.25em;">828</span><span style="margin-left:.25em;">459</span></span></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The natural logarithm of <span class="texhtml mvar" style="font-style:italic;">x</span> is generally written as <span class="texhtml">ln <i>x</i></span>, <span class="texhtml">log<sub><i>e</i></sub> <i>x</i></span>, or sometimes, if the base <span class="texhtml mvar" style="font-style:italic;">e</span> is implicit, simply <span class="texhtml">log <i>x</i></span>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Parentheses" class="mw-redirect" title="Parentheses">Parentheses</a> are sometimes added for clarity, giving <span class="texhtml">ln(<i>x</i>)</span>, <span class="texhtml">log<sub><i>e</i></sub>(<i>x</i>)</span>, or <span class="texhtml">log(<i>x</i>)</span>. This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. </p><p>The natural logarithm of <span class="texhtml mvar" style="font-style:italic;">x</span> is the <a href="/wiki/Exponentiation" title="Exponentiation">power</a> to which <span class="texhtml mvar" style="font-style:italic;">e</span> would have to be raised to equal <span class="texhtml mvar" style="font-style:italic;">x</span>. For example, <span class="texhtml">ln 7.5</span> is <span class="texhtml">2.0149...</span>, because <span class="texhtml"><i>e</i><sup>2.0149...</sup> = 7.5</span>. The natural logarithm of <span class="texhtml mvar" style="font-style:italic;">e</span> itself, <span class="texhtml">ln <i>e</i></span>, is <span class="texhtml">1</span>, because <span class="texhtml"><i>e</i><sup>1</sup> = <i>e</i></span>, while the natural logarithm of <span class="texhtml">1</span> is <span class="texhtml">0</span>, since <span class="texhtml"><i>e</i><sup>0</sup> = 1</span>. </p><p>The natural logarithm can be defined for any positive <a href="/wiki/Real_number" title="Real number">real number</a> <span class="texhtml mvar" style="font-style:italic;">a</span> as the <a href="/wiki/Integral" title="Integral">area under the curve</a> <span class="texhtml"><i>y</i> = 1/<i>x</i></span> from <span class="texhtml">1</span> to <span class="texhtml mvar" style="font-style:italic;">a</span><sup id="cite_ref-:1_4-0" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> (with the area being negative when <span class="texhtml">0 < <i>a</i> < 1</span>). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, although this leads to a <a href="/wiki/Multi-valued_function" class="mw-redirect" title="Multi-valued function">multi-valued function</a>: see <a href="/wiki/Complex_logarithm" title="Complex logarithm">complex logarithm</a> for more. </p><p>The natural logarithm function, if considered as a <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued function</a> of a positive real variable, is the <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> of the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, leading to the identities: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}e^{\ln x}&=x\qquad {\text{ if }}x\in \mathbb {R} _{+}\\\ln e^{x}&=x\qquad {\text{ if }}x\in \mathbb {R} \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> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> if </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{\ln x}&=x\qquad {\text{ if }}x\in \mathbb {R} _{+}\\\ln e^{x}&=x\qquad {\text{ if }}x\in \mathbb {R} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bea69d17f9d1fea1c15d5b0089ac7a49fbab8c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.286ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}e^{\ln x}&=x\qquad {\text{ if }}x\in \mathbb {R} _{+}\\\ln e^{x}&=x\qquad {\text{ if }}x\in \mathbb {R} \end{aligned}}}"></span> </p><p>Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:<sup id="cite_ref-:2_5-0" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x\cdot y)=\ln x+\ln y~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x\cdot y)=\ln x+\ln y~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d63556ff0cd798e432cf987fb621a309a1a7d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.217ex; height:2.843ex;" alt="{\displaystyle \ln(x\cdot y)=\ln x+\ln y~.}"></span> </p><p>Logarithms can be defined for any positive base other than 1, not only <span class="texhtml mvar" style="font-style:italic;">e</span>. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}x=\ln x/\ln b=\ln x\cdot \log _{b}e}"> <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> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>b</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>⋅</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}x=\ln x/\ln b=\ln x\cdot \log _{b}e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb0ac119bb0723ce83ec808a7ab3d5d16ed05a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.068ex; height:2.843ex;" alt="{\displaystyle \log _{b}x=\ln x/\ln b=\ln x\cdot \log _{b}e}"></span>. </p><p>Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the <a href="/wiki/Half-life" title="Half-life">half-life</a>, decay constant, or unknown time in <a href="/wiki/Exponential_decay" title="Exponential decay">exponential decay</a> problems. They are important in many branches of mathematics and scientific disciplines, and are used to solve problems involving <a href="/wiki/Compound_interest" title="Compound interest">compound interest</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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> <p>The concept of the natural logarithm was worked out by <a href="/wiki/Gregoire_de_Saint-Vincent" class="mw-redirect" title="Gregoire de Saint-Vincent">Gregoire de Saint-Vincent</a> and <a href="/wiki/Alphonse_Antonio_de_Sarasa" title="Alphonse Antonio de Sarasa">Alphonse Antonio de Sarasa</a> before 1649.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Their work involved <a href="/wiki/Quadrature_(mathematics)" class="mw-redirect" title="Quadrature (mathematics)">quadrature</a> of the <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> with equation <span class="texhtml"><i>xy</i> = 1</span>, by determination of the area of <a href="/wiki/Hyperbolic_sector" title="Hyperbolic sector">hyperbolic sectors</a>. Their solution generated the requisite "<a href="/wiki/Hyperbolic_logarithm" class="mw-redirect" title="Hyperbolic logarithm">hyperbolic logarithm</a>" <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, which had the properties now associated with the natural logarithm. </p><p>An early mention of the natural logarithm was by <a href="/wiki/Nicholas_Mercator" title="Nicholas Mercator">Nicholas Mercator</a> in his work <i>Logarithmotechnia</i>, published in 1668,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> although the mathematics teacher <a href="/wiki/John_Speidell" title="John Speidell">John Speidell</a> had already compiled a table of what in fact were effectively natural logarithms in 1619.<sup id="cite_ref-Cajori_8-0" class="reference"><a href="#cite_note-Cajori-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> It has been said that Speidell's logarithms were to the base <span class="texhtml mvar" style="font-style:italic;">e</span>, but this is not entirely true due to complications with the values being expressed as <a href="/wiki/Integer" title="Integer">integers</a>.<sup id="cite_ref-Cajori_8-1" class="reference"><a href="#cite_note-Cajori-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 152">: 152 </span></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Notational_conventions">Notational conventions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=2" title="Edit section: Notational conventions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The notations <span class="texhtml">ln <i>x</i></span> and <span class="texhtml">log<sub><i>e</i></sub> <i>x</i></span> both refer unambiguously to the natural logarithm of <span class="texhtml mvar" style="font-style:italic;">x</span>, and <span class="texhtml">log <i>x</i></span> without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many <a href="/wiki/Programming_language" title="Programming language">programming languages</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup> In some other contexts such as <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, however, <span class="texhtml">log <i>x</i></span> can be used to denote the <a href="/wiki/Common_logarithm" title="Common logarithm">common (base 10) logarithm</a>. It may also refer to the <a href="/wiki/Binary_logarithm" title="Binary logarithm">binary (base 2) logarithm</a> in the context of <a href="/wiki/Computer_science" title="Computer science">computer science</a>, particularly in the context of <a href="/wiki/Time_complexity" title="Time complexity">time complexity</a>. </p><p>Generally, the notation for the logarithm to base <span class="texhtml"><i>b</i></span> of a number <span class="texhtml"><i>x</i></span> is shown as <span class="texhtml">log<sub><i>b</i></sub> <i>x</i></span>. So the <span class="texhtml">log</span> of <span class="texhtml">8</span> to the base <span class="texhtml">2</span> would be <span class="texhtml">log<sub>2</sub> 8 = 3</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=3" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The natural logarithm can be defined in several equivalent ways. </p> <div class="mw-heading mw-heading3"><h3 id="Inverse_of_exponential">Inverse of exponential</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=4" title="Edit section: Inverse of exponential"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most general definition is as the inverse function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841c0d168e64191c45a45e54c7e447defd17ec6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}"></span>, so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{\ln(x)}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\ln(x)}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d124a7b0424b4ee111d26c87ca2027113f48408" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.335ex; height:2.843ex;" alt="{\displaystyle e^{\ln(x)}=x}"></span>. Because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841c0d168e64191c45a45e54c7e447defd17ec6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}"></span> is positive and invertible for any real input <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, this definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df055b8e294310e6785701c1c67105e109191d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \ln(x)}"></span> is well defined for any positive <span class="texhtml mvar" style="font-style:italic;">x</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Integral_definition">Integral definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=5" title="Edit section: Integral definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Log-pole-x_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Log-pole-x_1.svg/220px-Log-pole-x_1.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Log-pole-x_1.svg/330px-Log-pole-x_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Log-pole-x_1.svg/440px-Log-pole-x_1.svg.png 2x" data-file-width="601" data-file-height="301" /></a><figcaption><span class="texhtml">ln <i>a</i></span> as the area of the shaded region under the curve <span class="texhtml"><i>f</i>(<i>x</i>) = 1/<i>x</i></span> from <span class="texhtml">1</span> to <span class="texhtml mvar" style="font-style:italic;">a</span>. If <span class="texhtml mvar" style="font-style:italic;">a</span> is less than <span class="texhtml">1</span>, the area taken to be negative.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Log.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Log.gif/220px-Log.gif" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Log.gif/330px-Log.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Log.gif/440px-Log.gif 2x" data-file-width="632" data-file-height="472" /></a><figcaption>The area under the hyperbola satisfies the logarithm rule. Here <span class="texhtml"><i>A</i>(<i>s</i>,<i>t</i>)</span> denotes the area under the hyperbola between <span class="texhtml mvar" style="font-style:italic;">s</span> and <span class="texhtml mvar" style="font-style:italic;">t</span>.</figcaption></figure> <p>The natural logarithm of a positive, real number <span class="texhtml mvar" style="font-style:italic;">a</span> may be defined as the <a href="/wiki/Area" title="Area">area</a> under the graph of the <a href="/wiki/Hyperbola#Rectangular_hyperbola" title="Hyperbola">hyperbola</a> with equation <span class="texhtml"><i>y</i> = 1/<i>x</i></span> between <span class="texhtml"><i>x</i> = 1</span> and <span class="texhtml"><i>x</i> = <i>a</i></span>. This is the <a href="/wiki/Integral" title="Integral">integral</a><sup id="cite_ref-:1_4-1" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</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 a=\int _{1}^{a}{\frac {1}{x}}\,dx.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0643577384575f32c790c261f6bd4125c6a711c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.353ex; height:5.843ex;" alt="{\displaystyle \ln a=\int _{1}^{a}{\frac {1}{x}}\,dx.}"></span> If <span class="texhtml mvar" style="font-style:italic;">a</span> is in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c79c6838e423c1ed3c7ea532a56dc9f9dae8290b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}"></span>, then the region has <a href="/wiki/Negative_area" class="mw-redirect" title="Negative area">negative area</a>, and the logarithm is negative. </p><p>This function is a logarithm because it satisfies the fundamental multiplicative property of a logarithm:<sup id="cite_ref-:2_5-1" class="reference"><a href="#cite_note-:2-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(ab)=\ln a+\ln b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(ab)=\ln a+\ln b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54f573d66bb4926ef5a726edeb1c79c2ab37050" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.442ex; height:2.843ex;" alt="{\displaystyle \ln(ab)=\ln a+\ln b.}"></span> </p><p>This can be demonstrated by splitting the integral that defines <span class="texhtml">ln <i>ab</i></span> into two parts, and then making the <a href="/wiki/Integration_by_substitution" title="Integration by substitution">variable substitution</a> <span class="texhtml"><i>x</i> = <i>at</i></span> (so <span class="texhtml"><i>dx</i> = <i>a</i> <i>dt</i></span>) in the second part, as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln ab=\int _{1}^{ab}{\frac {1}{x}}\,dx&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{a}^{ab}{\frac {1}{x}}\,dx\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{at}}a\,dt\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{t}}\,dt\\[5pt]&=\ln a+\ln b.\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="0.8em 0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mi>b</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</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> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</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>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</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> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</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"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>a</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>a</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</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>a</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"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>b</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln ab=\int _{1}^{ab}{\frac {1}{x}}\,dx&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{a}^{ab}{\frac {1}{x}}\,dx\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{at}}a\,dt\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{t}}\,dt\\[5pt]&=\ln a+\ln b.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7210259ed243c3b86451e39eb2b50dccc7832e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.338ex; width:43.518ex; height:25.843ex;" alt="{\displaystyle {\begin{aligned}\ln ab=\int _{1}^{ab}{\frac {1}{x}}\,dx&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{a}^{ab}{\frac {1}{x}}\,dx\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{at}}a\,dt\\[5pt]&=\int _{1}^{a}{\frac {1}{x}}\,dx+\int _{1}^{b}{\frac {1}{t}}\,dt\\[5pt]&=\ln a+\ln b.\end{aligned}}}"></span> </p><p>In elementary terms, this is simply scaling by <span class="texhtml">1/<i>a</i></span> in the horizontal direction and by <span class="texhtml mvar" style="font-style:italic;">a</span> in the vertical direction. Area does not change under this transformation, but the region between <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml"><i>ab</i></span> is reconfigured. Because the function <span class="texhtml"><i>a</i>/(<i>ax</i>)</span> is equal to the function <span class="texhtml">1/<i>x</i></span>, the resulting area is precisely <span class="texhtml">ln <i>b</i></span>. </p><p>The number <span class="texhtml mvar" style="font-style:italic;">e</span> can then be defined to be the unique real number <span class="texhtml mvar" style="font-style:italic;">a</span> such that <span class="texhtml">ln <i>a</i> = 1</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=6" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The natural logarithm has the following mathematical properties: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln 1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln 1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b56677c77095e38669c0a4a5c7f885cee45faec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.75ex; height:2.176ex;" alt="{\displaystyle \ln 1=0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln e=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>e</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln e=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac39a99258627c071061334973932ca14bdf0a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.671ex; height:2.176ex;" alt="{\displaystyle \ln e=1}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(xy)=\ln x+\ln y\quad {\text{for }}\;x>0\;{\text{and }}\;y>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and </mtext> </mrow> <mspace width="thickmathspace" /> <mi>y</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(xy)=\ln x+\ln y\quad {\text{for }}\;x>0\;{\text{and }}\;y>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fadc8b5815591a884a92b90f72dc571da3a8db37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.271ex; height:2.843ex;" alt="{\displaystyle \ln(xy)=\ln x+\ln y\quad {\text{for }}\;x>0\;{\text{and }}\;y>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x/y)=\ln x-\ln y\quad {\text{for }}\;x>0\;{\text{and }}\;y>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and </mtext> </mrow> <mspace width="thickmathspace" /> <mi>y</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x/y)=\ln x-\ln y\quad {\text{for }}\;x>0\;{\text{and }}\;y>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e3745871727cdd8abc63b84e6784c0e043e2ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.433ex; height:2.843ex;" alt="{\displaystyle \ln(x/y)=\ln x-\ln y\quad {\text{for }}\;x>0\;{\text{and }}\;y>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x^{y})=y\ln x\quad {\text{for }}\;x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x^{y})=y\ln x\quad {\text{for }}\;x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525c7cfa5c3779af39c5f1ad063c7f1fef366490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.35ex; height:2.843ex;" alt="{\displaystyle \ln(x^{y})=y\ln x\quad {\text{for }}\;x>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln({\sqrt[{y}]{x}})=(\ln x)/y\quad {\text{for }}\;x>0\;{\text{and }}\;y\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and </mtext> </mrow> <mspace width="thickmathspace" /> <mi>y</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln({\sqrt[{y}]{x}})=(\ln x)/y\quad {\text{for }}\;x>0\;{\text{and }}\;y\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/196be723b66e16fed1cf00d43cc450d2e93468d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.856ex; height:3.009ex;" alt="{\displaystyle \ln({\sqrt[{y}]{x}})=(\ln x)/y\quad {\text{for }}\;x>0\;{\text{and }}\;y\neq 0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x<\ln y\quad {\text{for }}\;0<x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo><</mo> <mi>ln</mi> <mo>⁡</mo> <mi>y</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mspace width="thickmathspace" /> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x<\ln y\quad {\text{for }}\;0<x<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e89c03159c1868263698c9a3500147aa8378272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.415ex; height:2.509ex;" alt="{\displaystyle \ln x<\ln y\quad {\text{for }}\;0<x<y}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a32ef3d5fbd5df0406c68a2ad58e415ae641d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.971ex; height:5.843ex;" alt="{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\alpha \to 0}{\frac {x^{\alpha }-1}{\alpha }}=\ln x\quad {\text{for }}\;x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>α</mi> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\alpha \to 0}{\frac {x^{\alpha }-1}{\alpha }}=\ln x\quad {\text{for }}\;x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffdf20e3bd949f7095ade82873d87fdfbf5150dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.037ex; height:5.343ex;" alt="{\displaystyle \lim _{\alpha \to 0}{\frac {x^{\alpha }-1}{\alpha }}=\ln x\quad {\text{for }}\;x>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1\quad {\text{for}}\quad x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>≤</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>≤</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1\quad {\text{for}}\quad x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd8d97775e47c1771dc9ddddf7d14107f07045b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.376ex; height:5.176ex;" alt="{\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1\quad {\text{for}}\quad x>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x\quad {\text{for}}\quad x\geq 0\;{\text{and }}\;\alpha \geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>α</mi> <mi>x</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>≥</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and </mtext> </mrow> <mspace width="thickmathspace" /> <mi>α</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x\quad {\text{for}}\quad x\geq 0\;{\text{and }}\;\alpha \geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/830dffb6e8a1ac98bddc12a646c57c047ac15378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.057ex; height:2.843ex;" alt="{\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x\quad {\text{for}}\quad x\geq 0\;{\text{and }}\;\alpha \geq 1}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Derivative">Derivative</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=7" title="Edit section: Derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Derivative" title="Derivative">derivative</a> of the natural logarithm as a <a href="/wiki/Real-valued_function" title="Real-valued function">real-valued function</a> on the positive reals is given by<sup id="cite_ref-:1_4-2" class="reference"><a href="#cite_note-:1-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\ln x={\frac {1}{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>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\ln x={\frac {1}{x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/908d58e860196c61db8f688876e2671bf1c812b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.336ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\ln x={\frac {1}{x}}.}"></span> </p><p>How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac5b5f4ac44fb3c5a6197256646ff2e8470bce7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.866ex; height:5.843ex;" alt="{\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,}"></span> then the derivative immediately follows from the first part of the <a href="/wiki/Fundamental_theorem_of_calculus#First_part" title="Fundamental theorem of calculus">fundamental theorem of calculus</a>. </p><p>On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for <span class="texhtml"><i>x</i> > 0</span>) can be found by using the properties of the logarithm and a definition of the exponential function. </p><p>From the definition of the number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=\lim _{u\to 0}(1+u)^{1/u},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>u</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=\lim _{u\to 0}(1+u)^{1/u},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26d4e219397ac4799a152c8e696f2a133c748ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.192ex; height:4.509ex;" alt="{\displaystyle e=\lim _{u\to 0}(1+u)^{1/u},}"></span> the exponential function can be defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}=\lim _{u\to 0}(1+u)^{x/u}=\lim _{h\to 0}(1+hx)^{1/h},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>u</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>h</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>h</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}=\lim _{u\to 0}(1+u)^{x/u}=\lim _{h\to 0}(1+hx)^{1/h},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51d090eb18b5669a945be91207d203b5c689bd5d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.297ex; height:4.509ex;" alt="{\displaystyle e^{x}=\lim _{u\to 0}(1+u)^{x/u}=\lim _{h\to 0}(1+hx)^{1/h},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=hx,h={\frac {u}{x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>h</mi> <mi>x</mi> <mo>,</mo> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mi>x</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=hx,h={\frac {u}{x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1801253a28423a270e1f423296f273914a624eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.381ex; height:4.676ex;" alt="{\displaystyle u=hx,h={\frac {u}{x}}.}"></span> </p><p>The derivative can then be found from first principles. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\ln x&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}\left[{\frac {1}{h}}\ln \left({\frac {x+h}{x}}\right)\right]\\&=\lim _{h\to 0}\left[\ln \left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{all above for logarithmic properties}}\\&=\ln \left[\lim _{h\to 0}\left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{for continuity of the logarithm}}\\&=\ln e^{1/x}\quad &&{\text{for the definition of }}e^{x}=\lim _{h\to 0}(1+hx)^{1/h}\\&={\frac {1}{x}}\quad &&{\text{for the definition of the ln as inverse function.}}\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> <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>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mi>h</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>h</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mspace width="1em" /> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>all above for logarithmic properties</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mspace width="1em" /> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for continuity of the logarithm</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mrow> </msup> <mspace width="1em" /> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for the definition of </mtext> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>h</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>h</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="1em" /> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>for the definition of the ln as inverse function.</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {d}{dx}}\ln x&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}\left[{\frac {1}{h}}\ln \left({\frac {x+h}{x}}\right)\right]\\&=\lim _{h\to 0}\left[\ln \left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{all above for logarithmic properties}}\\&=\ln \left[\lim _{h\to 0}\left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{for continuity of the logarithm}}\\&=\ln e^{1/x}\quad &&{\text{for the definition of }}e^{x}=\lim _{h\to 0}(1+hx)^{1/h}\\&={\frac {1}{x}}\quad &&{\text{for the definition of the ln as inverse function.}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e12dece20904dd40cd69415a900d0ee66e36793" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.005ex; width:84.359ex; height:39.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {d}{dx}}\ln x&=\lim _{h\to 0}{\frac {\ln(x+h)-\ln x}{h}}\\&=\lim _{h\to 0}\left[{\frac {1}{h}}\ln \left({\frac {x+h}{x}}\right)\right]\\&=\lim _{h\to 0}\left[\ln \left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{all above for logarithmic properties}}\\&=\ln \left[\lim _{h\to 0}\left(1+{\frac {h}{x}}\right)^{\frac {1}{h}}\right]\quad &&{\text{for continuity of the logarithm}}\\&=\ln e^{1/x}\quad &&{\text{for the definition of }}e^{x}=\lim _{h\to 0}(1+hx)^{1/h}\\&={\frac {1}{x}}\quad &&{\text{for the definition of the ln as inverse function.}}\end{aligned}}}"></span> </p><p>Also, we have: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\ln ax={\frac {d}{dx}}(\ln a+\ln x)={\frac {d}{dx}}\ln a+{\frac {d}{dx}}\ln x={\frac {1}{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>a</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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>a</mi> <mo>+</mo> <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>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\ln ax={\frac {d}{dx}}(\ln a+\ln x)={\frac {d}{dx}}\ln a+{\frac {d}{dx}}\ln x={\frac {1}{x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b3d69920eaa95c5c0e6303a5a7887b25f96d94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.597ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\ln ax={\frac {d}{dx}}(\ln a+\ln x)={\frac {d}{dx}}\ln a+{\frac {d}{dx}}\ln x={\frac {1}{x}}.}"></span> </p><p>so, unlike its inverse function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ax}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{ax}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c922caffb8d5be4d157a315ce579532f2392e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.126ex; height:2.343ex;" alt="{\displaystyle e^{ax}}"></span>, a constant in the function doesn't alter the differential. </p> <div class="mw-heading mw-heading2"><h2 id="Series">Series</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=8" title="Edit section: Series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LogTay.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/LogTay.svg/290px-LogTay.svg.png" decoding="async" width="290" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/LogTay.svg/435px-LogTay.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/LogTay.svg/580px-LogTay.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>The Taylor polynomials for <span class="texhtml">ln(1 + <i>x</i>)</span> only provide accurate approximations in the range <span class="texhtml">−1 < <i>x</i> ≤ 1</span>. Beyond some <span class="texhtml"><i>x</i> > 1</span>, the Taylor polynomials of higher degree are increasingly <i>worse</i> approximations.</figcaption></figure> <p>Since the natural logarithm is undefined at 0, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df055b8e294310e6785701c1c67105e109191d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \ln(x)}"></span> itself does not have a <a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a>, unlike many other elementary functions. Instead, one looks for Taylor expansions around other points. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert x-1\vert \leq 1{\text{ and }}x\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">|</mo> <mo>≤</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert x-1\vert \leq 1{\text{ and }}x\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62594fa9e6d41d0435b13d1cb7b5a2249bf0584d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.033ex; height:2.843ex;" alt="{\displaystyle \vert x-1\vert \leq 1{\text{ and }}x\neq 0,}"></span> then<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln x&=\int _{1}^{x}{\frac {1}{t}}\,dt=\int _{0}^{x-1}{\frac {1}{1+u}}\,du\\&=\int _{0}^{x-1}(1-u+u^{2}-u^{3}+\cdots )\,du\\&=(x-1)-{\frac {(x-1)^{2}}{2}}+{\frac {(x-1)^{3}}{3}}-{\frac {(x-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(x-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> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>u</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>u</mi> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</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>x</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>x</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <mo stretchy="false">(</mo> <mi>x</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 x&=\int _{1}^{x}{\frac {1}{t}}\,dt=\int _{0}^{x-1}{\frac {1}{1+u}}\,du\\&=\int _{0}^{x-1}(1-u+u^{2}-u^{3}+\cdots )\,du\\&=(x-1)-{\frac {(x-1)^{2}}{2}}+{\frac {(x-1)^{3}}{3}}-{\frac {(x-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(x-1)^{k}}{k}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858d55fd9e485796f4404f743a9f22a8d6c248b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.505ex; width:55.829ex; height:26.176ex;" alt="{\displaystyle {\begin{aligned}\ln x&=\int _{1}^{x}{\frac {1}{t}}\,dt=\int _{0}^{x-1}{\frac {1}{1+u}}\,du\\&=\int _{0}^{x-1}(1-u+u^{2}-u^{3}+\cdots )\,du\\&=(x-1)-{\frac {(x-1)^{2}}{2}}+{\frac {(x-1)^{3}}{3}}-{\frac {(x-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}(x-1)^{k}}{k}}.\end{aligned}}}"></span> </p><p>This is the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed172b0f5195382a3500c713941f945ad4db3898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \ln x}"></span> around 1. A change of variables yields the <a href="/wiki/Mercator_series" title="Mercator series">Mercator series</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots ,}"> <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>x</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> <mrow> <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> <mi>k</mi> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8e0fad95d6522197dd0c46c0e3f7eb20266aec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.292ex; height:7.009ex;" alt="{\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots ,}"></span> valid for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|\leq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad95e9e3840f09ecffbe53c104ee1bbf639fd42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |x|\leq 1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq -1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq -1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d03e6133a3fc9f8e910ad3002e65f6d1025dc37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.046ex; height:2.676ex;" alt="{\displaystyle x\neq -1.}"></span> </p><p><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> disregarding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\neq -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a83e8331c458b7378b08c5ef4cce193113e68e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.399ex; height:2.676ex;" alt="{\displaystyle x\neq -1}"></span>, nevertheless applied this series to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fefa55268918f98da2e0dcc19ea86d78f84ac56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x=-1}"></span> to show that the <a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">harmonic series</a> equals the natural logarithm of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/314d849898dc01e1117caf96dd9318d4e67ff194" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.001ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{1-1}}}"></span>; that is, the logarithm of infinity. Nowadays, more formally, one can prove that the harmonic series truncated at <span class="texhtml mvar" style="font-style:italic;">N</span> is close to the logarithm of <span class="texhtml mvar" style="font-style:italic;">N</span>, when <span class="texhtml mvar" style="font-style:italic;">N</span> is large, with the difference converging to the <a href="/wiki/Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>. </p><p>The figure is a <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of <span class="texhtml">ln(1 + <i>x</i>)</span> and some of its <a href="/wiki/Taylor_polynomial" class="mw-redirect" title="Taylor polynomial">Taylor polynomials</a> around 0. These approximations converge to the function only in the region <span class="texhtml">−1 < <i>x</i> ≤ 1</span>; outside this region, the higher-degree Taylor polynomials devolve to <i>worse</i> approximations for the function. </p><p>A useful special case for positive integers <span class="texhtml mvar" style="font-style:italic;">n</span>, taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fbd81bf9cfb2df19ef99f4522e6607e3afed278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.25ex; height:3.343ex;" alt="{\displaystyle x={\tfrac {1}{n}}}"></span>, is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{kn^{k}}}={\frac {1}{n}}-{\frac {1}{2n^{2}}}+{\frac {1}{3n^{3}}}-{\frac {1}{4n^{4}}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </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 mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <mrow> <mi>k</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{kn^{k}}}={\frac {1}{n}}-{\frac {1}{2n^{2}}}+{\frac {1}{3n^{3}}}-{\frac {1}{4n^{4}}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e7515183c5d10c12292820a9e22b9c0d43ff66" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.997ex; height:7.009ex;" alt="{\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{kn^{k}}}={\frac {1}{n}}-{\frac {1}{2n^{2}}}+{\frac {1}{3n^{3}}}-{\frac {1}{4n^{4}}}+\cdots }"></span> </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (x)\geq 1/2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (x)\geq 1/2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb2ec93ff18738aafe3556fb38f70896136dad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.115ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (x)\geq 1/2,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln(x)&=-\ln \left({\frac {1}{x}}\right)=-\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}({\frac {1}{x}}-1)^{k}}{k}}=\sum _{k=1}^{\infty }{\frac {(x-1)^{k}}{kx^{k}}}\\&={\frac {x-1}{x}}+{\frac {(x-1)^{2}}{2x^{2}}}+{\frac {(x-1)^{3}}{3x^{3}}}+{\frac {(x-1)^{4}}{4x^{4}}}+\cdots \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>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</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> <mrow> <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> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <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> <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> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mi>k</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln(x)&=-\ln \left({\frac {1}{x}}\right)=-\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}({\frac {1}{x}}-1)^{k}}{k}}=\sum _{k=1}^{\infty }{\frac {(x-1)^{k}}{kx^{k}}}\\&={\frac {x-1}{x}}+{\frac {(x-1)^{2}}{2x^{2}}}+{\frac {(x-1)^{3}}{3x^{3}}}+{\frac {(x-1)^{4}}{4x^{4}}}+\cdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2102e05f53bf82c164e2c7d43c26eb6092c93c75" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:61.075ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}\ln(x)&=-\ln \left({\frac {1}{x}}\right)=-\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}({\frac {1}{x}}-1)^{k}}{k}}=\sum _{k=1}^{\infty }{\frac {(x-1)^{k}}{kx^{k}}}\\&={\frac {x-1}{x}}+{\frac {(x-1)^{2}}{2x^{2}}}+{\frac {(x-1)^{3}}{3x^{3}}}+{\frac {(x-1)^{4}}{4x^{4}}}+\cdots \end{aligned}}}"></span> </p><p>Now, taking <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {n+1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {n+1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66dec579589f5c96e5f9e11e24d0de2e63826b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.351ex; height:3.509ex;" alt="{\displaystyle x={\tfrac {n+1}{n}}}"></span> for positive integers <span class="texhtml mvar" style="font-style:italic;">n</span>, we get: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {1}{k(n+1)^{k}}}={\frac {1}{n+1}}+{\frac {1}{2(n+1)^{2}}}+{\frac {1}{3(n+1)^{3}}}+{\frac {1}{4(n+1)^{4}}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </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 mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {1}{k(n+1)^{k}}}={\frac {1}{n+1}}+{\frac {1}{2(n+1)^{2}}}+{\frac {1}{3(n+1)^{3}}}+{\frac {1}{4(n+1)^{4}}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1991bf28c2e7b3e3ebe24a87d25959c20fe8e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:82.974ex; height:6.843ex;" alt="{\displaystyle \ln \left({\frac {n+1}{n}}\right)=\sum _{k=1}^{\infty }{\frac {1}{k(n+1)^{k}}}={\frac {1}{n+1}}+{\frac {1}{2(n+1)^{2}}}+{\frac {1}{3(n+1)^{3}}}+{\frac {1}{4(n+1)^{4}}}+\cdots }"></span> </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (x)\geq 0{\text{ and }}x\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Re</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (x)\geq 0{\text{ and }}x\neq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418695d432e5bb7ec6e53e46d568ca7c1952bd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.289ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (x)\geq 0{\text{ and }}x\neq 0,}"></span> then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)=\ln \left({\frac {2x}{2}}\right)=\ln \left({\frac {1+{\frac {x-1}{x+1}}}{1-{\frac {x-1}{x+1}}}}\right)=\ln \left(1+{\frac {x-1}{x+1}}\right)-\ln \left(1-{\frac {x-1}{x+1}}\right).}"> <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> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)=\ln \left({\frac {2x}{2}}\right)=\ln \left({\frac {1+{\frac {x-1}{x+1}}}{1-{\frac {x-1}{x+1}}}}\right)=\ln \left(1+{\frac {x-1}{x+1}}\right)-\ln \left(1-{\frac {x-1}{x+1}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcee9f0c0c50b381d8e67e3596f332f09b4882c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:72.336ex; height:8.509ex;" alt="{\displaystyle \ln(x)=\ln \left({\frac {2x}{2}}\right)=\ln \left({\frac {1+{\frac {x-1}{x+1}}}{1-{\frac {x-1}{x+1}}}}\right)=\ln \left(1+{\frac {x-1}{x+1}}\right)-\ln \left(1-{\frac {x-1}{x+1}}\right).}"></span> Since <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln(1+y)-\ln(1-y)&=\sum _{i=1}^{\infty }{\frac {1}{i}}\left((-1)^{i-1}y^{i}-(-1)^{i-1}(-y)^{i}\right)=\sum _{i=1}^{\infty }{\frac {y^{i}}{i}}\left((-1)^{i-1}+1\right)\\&=y\sum _{i=1}^{\infty }{\frac {y^{i-1}}{i}}\left((-1)^{i-1}+1\right){\overset {i-1\to 2k}{=}}\;2y\sum _{k=0}^{\infty }{\frac {y^{2k}}{2k+1}},\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> <mn>1</mn> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>i</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</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>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>i</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>y</mi> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</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>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>i</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>=</mo> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">→</mo> <mn>2</mn> <mi>k</mi> </mrow> </mover> </mrow> <mspace width="thickmathspace" /> <mn>2</mn> <mi>y</mi> <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> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln(1+y)-\ln(1-y)&=\sum _{i=1}^{\infty }{\frac {1}{i}}\left((-1)^{i-1}y^{i}-(-1)^{i-1}(-y)^{i}\right)=\sum _{i=1}^{\infty }{\frac {y^{i}}{i}}\left((-1)^{i-1}+1\right)\\&=y\sum _{i=1}^{\infty }{\frac {y^{i-1}}{i}}\left((-1)^{i-1}+1\right){\overset {i-1\to 2k}{=}}\;2y\sum _{k=0}^{\infty }{\frac {y^{2k}}{2k+1}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6d6827e88d696c901f9d6dc4124b4b393dec82" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.233ex; margin-bottom: -0.272ex; width:82.332ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\ln(1+y)-\ln(1-y)&=\sum _{i=1}^{\infty }{\frac {1}{i}}\left((-1)^{i-1}y^{i}-(-1)^{i-1}(-y)^{i}\right)=\sum _{i=1}^{\infty }{\frac {y^{i}}{i}}\left((-1)^{i-1}+1\right)\\&=y\sum _{i=1}^{\infty }{\frac {y^{i-1}}{i}}\left((-1)^{i-1}+1\right){\overset {i-1\to 2k}{=}}\;2y\sum _{k=0}^{\infty }{\frac {y^{2k}}{2k+1}},\end{aligned}}}"></span> we arrive at <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln(x)&={\frac {2(x-1)}{x+1}}\sum _{k=0}^{\infty }{\frac {1}{2k+1}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{k}\\&={\frac {2(x-1)}{x+1}}\left({\frac {1}{1}}+{\frac {1}{3}}{\frac {(x-1)^{2}}{(x+1)^{2}}}+{\frac {1}{5}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{2}+\cdots \right).\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>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <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 class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <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> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln(x)&={\frac {2(x-1)}{x+1}}\sum _{k=0}^{\infty }{\frac {1}{2k+1}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{k}\\&={\frac {2(x-1)}{x+1}}\left({\frac {1}{1}}+{\frac {1}{3}}{\frac {(x-1)^{2}}{(x+1)^{2}}}+{\frac {1}{5}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{2}+\cdots \right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea9e55f4830588e9d009067861916160bac91f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.671ex; width:63.597ex; height:16.509ex;" alt="{\displaystyle {\begin{aligned}\ln(x)&={\frac {2(x-1)}{x+1}}\sum _{k=0}^{\infty }{\frac {1}{2k+1}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{k}\\&={\frac {2(x-1)}{x+1}}\left({\frac {1}{1}}+{\frac {1}{3}}{\frac {(x-1)^{2}}{(x+1)^{2}}}+{\frac {1}{5}}{\left({\frac {(x-1)^{2}}{(x+1)^{2}}}\right)}^{2}+\cdots \right).\end{aligned}}}"></span> Using the substitution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {n+1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {n+1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66dec579589f5c96e5f9e11e24d0de2e63826b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.351ex; height:3.509ex;" alt="{\displaystyle x={\tfrac {n+1}{n}}}"></span> again for positive integers <span class="texhtml mvar" style="font-style:italic;">n</span>, we get: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln \left({\frac {n+1}{n}}\right)&={\frac {2}{2n+1}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)((2n+1)^{2})^{k}}}\\&=2\left({\frac {1}{2n+1}}+{\frac {1}{3(2n+1)^{3}}}+{\frac {1}{5(2n+1)^{5}}}+\cdots \right).\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> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <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> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <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 class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>5</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln \left({\frac {n+1}{n}}\right)&={\frac {2}{2n+1}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)((2n+1)^{2})^{k}}}\\&=2\left({\frac {1}{2n+1}}+{\frac {1}{3(2n+1)^{3}}}+{\frac {1}{5(2n+1)^{5}}}+\cdots \right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5310011cce8c49f30329e2cdcb46f17ec05ec70c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:62.934ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\ln \left({\frac {n+1}{n}}\right)&={\frac {2}{2n+1}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)((2n+1)^{2})^{k}}}\\&=2\left({\frac {1}{2n+1}}+{\frac {1}{3(2n+1)^{3}}}+{\frac {1}{5(2n+1)^{5}}}+\cdots \right).\end{aligned}}}"></span> </p><p>This is, by far, the fastest converging of the series described here. </p><p>The natural logarithm can also be expressed as an infinite product:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)=(x-1)\prod _{k=1}^{\infty }\left({\frac {2}{1+{\sqrt[{2^{k}}]{x}}}}\right)}"> <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> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</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> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)=(x-1)\prod _{k=1}^{\infty }\left({\frac {2}{1+{\sqrt[{2^{k}}]{x}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a685fc50f560fbfd45cd8c625c839137a0cb42" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.086ex; height:7.509ex;" alt="{\displaystyle \ln(x)=(x-1)\prod _{k=1}^{\infty }\left({\frac {2}{1+{\sqrt[{2^{k}}]{x}}}}\right)}"></span> </p><p>Two examples might be: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(2)=\left({\frac {2}{1+{\sqrt {2}}}}\right)\left({\frac {2}{1+{\sqrt[{4}]{2}}}}\right)\left({\frac {2}{1+{\sqrt[{8}]{2}}}}\right)\left({\frac {2}{1+{\sqrt[{16}]{2}}}}\right)...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(2)=\left({\frac {2}{1+{\sqrt {2}}}}\right)\left({\frac {2}{1+{\sqrt[{4}]{2}}}}\right)\left({\frac {2}{1+{\sqrt[{8}]{2}}}}\right)\left({\frac {2}{1+{\sqrt[{16}]{2}}}}\right)...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b509b3f7ce72ef554df0dfb48f10dbdc772055" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.505ex; height:7.509ex;" alt="{\displaystyle \ln(2)=\left({\frac {2}{1+{\sqrt {2}}}}\right)\left({\frac {2}{1+{\sqrt[{4}]{2}}}}\right)\left({\frac {2}{1+{\sqrt[{8}]{2}}}}\right)\left({\frac {2}{1+{\sqrt[{16}]{2}}}}\right)...}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =(2i+2)\left({\frac {2}{1+{\sqrt {i}}}}\right)\left({\frac {2}{1+{\sqrt[{4}]{i}}}}\right)\left({\frac {2}{1+{\sqrt[{8}]{i}}}}\right)\left({\frac {2}{1+{\sqrt[{16}]{i}}}}\right)...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>i</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =(2i+2)\left({\frac {2}{1+{\sqrt {i}}}}\right)\left({\frac {2}{1+{\sqrt[{4}]{i}}}}\right)\left({\frac {2}{1+{\sqrt[{8}]{i}}}}\right)\left({\frac {2}{1+{\sqrt[{16}]{i}}}}\right)...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/438561a80d9075271b2dcd15bea94412d2683e69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.65ex; height:7.509ex;" alt="{\displaystyle \pi =(2i+2)\left({\frac {2}{1+{\sqrt {i}}}}\right)\left({\frac {2}{1+{\sqrt[{4}]{i}}}}\right)\left({\frac {2}{1+{\sqrt[{8}]{i}}}}\right)\left({\frac {2}{1+{\sqrt[{16}]{i}}}}\right)...}"></span> </p><p>From this identity, we can easily get that: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\ln(x)}}={\frac {x}{x-1}}-\sum _{k=1}^{\infty }{\frac {2^{-k}x^{2^{-k}}}{1+x^{2^{-k}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </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 mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\ln(x)}}={\frac {x}{x-1}}-\sum _{k=1}^{\infty }{\frac {2^{-k}x^{2^{-k}}}{1+x^{2^{-k}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40e05f33901dca82a4dcbd75c62e5608103fb4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.885ex; height:7.176ex;" alt="{\displaystyle {\frac {1}{\ln(x)}}={\frac {x}{x-1}}-\sum _{k=1}^{\infty }{\frac {2^{-k}x^{2^{-k}}}{1+x^{2^{-k}}}}}"></span> </p><p>For example: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\ln(2)}}=2-{\frac {\sqrt {2}}{2+2{\sqrt {2}}}}-{\frac {\sqrt[{4}]{2}}{4+4{\sqrt[{4}]{2}}}}-{\frac {\sqrt[{8}]{2}}{8+8{\sqrt[{8}]{2}}}}\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mrow> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> <mrow> <mn>4</mn> <mo>+</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mroot> <mrow> <mn>8</mn> <mo>+</mo> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mroot> </mrow> </mrow> </mfrac> </mrow> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\ln(2)}}=2-{\frac {\sqrt {2}}{2+2{\sqrt {2}}}}-{\frac {\sqrt[{4}]{2}}{4+4{\sqrt[{4}]{2}}}}-{\frac {\sqrt[{8}]{2}}{8+8{\sqrt[{8}]{2}}}}\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/def29e93c87165f8547949c7e10896c45dc22d2f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:48.939ex; height:6.843ex;" alt="{\displaystyle {\frac {1}{\ln(2)}}=2-{\frac {\sqrt {2}}{2+2{\sqrt {2}}}}-{\frac {\sqrt[{4}]{2}}{4+4{\sqrt[{4}]{2}}}}-{\frac {\sqrt[{8}]{2}}{8+8{\sqrt[{8}]{2}}}}\cdots }"></span> </p> <div class="mw-heading mw-heading2"><h2 id="The_natural_logarithm_in_integration">The natural logarithm in integration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=9" title="Edit section: The natural logarithm in integration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The natural logarithm allows simple <a href="/wiki/Integral" title="Integral">integration</a> of functions of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)={\frac {f'(x)}{f(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)={\frac {f'(x)}{f(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/855ec85038d18bd7c78c473575ac4191ecac341b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.334ex; height:6.509ex;" alt="{\displaystyle g(x)={\frac {f'(x)}{f(x)}}}"></span>: an <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> of <span class="texhtml"><i>g</i>(<i>x</i>)</span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(|f(x)|)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(|f(x)|)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2d8844c090915cbf322d60fb3db8334c0715c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.46ex; height:2.843ex;" alt="{\displaystyle \ln(|f(x)|)}"></span>. This is the case because of the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> and the following fact: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\ln \left|x\right|={\frac {1}{x}},\ \ x\neq 0}"> <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> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\ln \left|x\right|={\frac {1}{x}},\ \ x\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae8029394c7708283e45924d8db2c688b165c79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.382ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}\ln \left|x\right|={\frac {1}{x}},\ \ x\neq 0}"></span> </p><p>In other words, when integrating over an interval of the real line that does not include <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <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> <mi>ln</mi> <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> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6503f208923d2c32e8adf60f9ed2871a1bd11fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.334ex; height:5.676ex;" alt="{\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|+C}"></span> where <span class="texhtml mvar" style="font-style:italic;">C</span> is an <a href="/wiki/Arbitrary_constant_of_integration" class="mw-redirect" title="Arbitrary constant of integration">arbitrary constant of integration</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Likewise, when the integral is over an interval where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2d21ef9ceb5237a7dc69a512c2d38a5e351e0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)\neq 0}"></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {{\frac {f'(x)}{f(x)}}\,dx}=\ln |f(x)|+C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <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> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {{\frac {f'(x)}{f(x)}}\,dx}=\ln |f(x)|+C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2683aa09c111fd31ccc4bc0759816701d790f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.884ex; height:6.509ex;" alt="{\displaystyle \int {{\frac {f'(x)}{f(x)}}\,dx}=\ln |f(x)|+C.}"></span></dd></dl> <p>For example, consider the integral of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398da5ded10e1ab022cfc8c3f4a4a87b46cd8c46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.499ex; height:2.843ex;" alt="{\displaystyle \tan(x)}"></span> over an interval that does not include points where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398da5ded10e1ab022cfc8c3f4a4a87b46cd8c46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.499ex; height:2.843ex;" alt="{\displaystyle \tan(x)}"></span> is infinite: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \tan x\,dx=\int {\frac {\sin x}{\cos x}}\,dx=-\int {\frac {{\frac {d}{dx}}\cos x}{\cos x}}\,dx=-\ln \left|\cos x\right|+C=\ln \left|\sec x\right|+C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mrow> <mi>sec</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \tan x\,dx=\int {\frac {\sin x}{\cos x}}\,dx=-\int {\frac {{\frac {d}{dx}}\cos x}{\cos x}}\,dx=-\ln \left|\cos x\right|+C=\ln \left|\sec x\right|+C.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ca4fc154b4954a63d6a437c8a1394db797ab1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:78.604ex; height:7.009ex;" alt="{\displaystyle \int \tan x\,dx=\int {\frac {\sin x}{\cos x}}\,dx=-\int {\frac {{\frac {d}{dx}}\cos x}{\cos x}}\,dx=-\ln \left|\cos x\right|+C=\ln \left|\sec x\right|+C.}"></span> </p><p>The natural logarithm can be integrated using <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><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> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a135b55276b11a6e857a789cfdddd890ebbebc00" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.065ex; height:5.676ex;" alt="{\displaystyle \int \ln x\,dx=x\ln x-x+C.}"></span> </p><p>Let: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=\ln x\Rightarrow du={\frac {dx}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo stretchy="false">⇒</mo> <mi>d</mi> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=\ln x\Rightarrow du={\frac {dx}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c39102e5ba5337bc67c2df0e1c53eca8197f92" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.724ex; height:5.343ex;" alt="{\displaystyle u=\ln x\Rightarrow du={\frac {dx}{x}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dv=dx\Rightarrow v=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>v</mi> <mo>=</mo> <mi>d</mi> <mi>x</mi> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dv=dx\Rightarrow v=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a87648e21c94fcadec9e9e1b30a3524f049ba2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.157ex; height:2.176ex;" alt="{\displaystyle dv=dx\Rightarrow v=x}"></span> then: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int \ln x\,dx&=x\ln x-\int {\frac {x}{x}}\,dx\\&=x\ln x-\int 1\,dx\\&=x\ln x-x+C\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> <mo>∫</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int \ln x\,dx&=x\ln x-\int {\frac {x}{x}}\,dx\\&=x\ln x-\int 1\,dx\\&=x\ln x-x+C\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adeb2b2f7fe0e299c2094658ae4b3f45772a4f3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:28.912ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\int \ln x\,dx&=x\ln x-\int {\frac {x}{x}}\,dx\\&=x\ln x-\int 1\,dx\\&=x\ln x-x+C\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Efficient_computation">Efficient computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=10" title="Edit section: Efficient computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df055b8e294310e6785701c1c67105e109191d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \ln(x)}"></span> where <span class="texhtml"><i>x</i> > 1</span>, the closer the value of <span class="texhtml mvar" style="font-style:italic;">x</span> is to 1, the faster the rate of convergence of its Taylor series centered at 1. The identities associated with the logarithm can be leveraged to exploit this: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln 123.456&=\ln(1.23456\cdot 10^{2})\\&=\ln 1.23456+\ln(10^{2})\\&=\ln 1.23456+2\ln 10\\&\approx \ln 1.23456+2\cdot 2.3025851.\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> <mn>123.456</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1.23456</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mn>1.23456</mn> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mn>1.23456</mn> <mo>+</mo> <mn>2</mn> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mi>ln</mi> <mo>⁡</mo> <mn>1.23456</mn> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mn>2.3025851.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln 123.456&=\ln(1.23456\cdot 10^{2})\\&=\ln 1.23456+\ln(10^{2})\\&=\ln 1.23456+2\ln 10\\&\approx \ln 1.23456+2\cdot 2.3025851.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768197b64aa3d694436082e076a44d335c8ece48" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:40.021ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\ln 123.456&=\ln(1.23456\cdot 10^{2})\\&=\ln 1.23456+\ln(10^{2})\\&=\ln 1.23456+2\ln 10\\&\approx \ln 1.23456+2\cdot 2.3025851.\end{aligned}}}"></span> </p><p>Such techniques were used before calculators, by referring to numerical tables and performing manipulations such as those above. </p> <div class="mw-heading mw-heading3"><h3 id="Natural_logarithm_of_10">Natural logarithm of 10</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=11" title="Edit section: Natural logarithm of 10"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The natural logarithm of 10, approximately equal to <span class="texhtml"><span class="nowrap"><span data-sort-value="7000230258509000000♠"></span>2.302<span style="margin-left:.25em;">585</span><span style="margin-left:.25em;">09</span></span></span>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> plays a role for example in the computation of natural logarithms of numbers represented in <a href="/wiki/Scientific_notation" title="Scientific notation">scientific notation</a>, as a <a href="/wiki/Mantissa_(logarithm)" class="mw-redirect" title="Mantissa (logarithm)">mantissa</a> multiplied by a power of 10: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(a\cdot 10^{n})=\ln a+n\ln 10.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mi>a</mi> <mo>+</mo> <mi>n</mi> <mi>ln</mi> <mo>⁡</mo> <mn>10.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(a\cdot 10^{n})=\ln a+n\ln 10.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4eb34e07344c3c4fb315fe7d937673afc038f5e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.776ex; height:2.843ex;" alt="{\displaystyle \ln(a\cdot 10^{n})=\ln a+n\ln 10.}"></span> </p><p>This means that one can effectively calculate the logarithms of numbers with very large or very small <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> using the logarithms of a relatively small set of decimals in the range <span class="texhtml">[1, 10)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="High_precision"><span class="anchor" id="lnp1"></span>High precision</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=12" title="Edit section: High precision"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To compute the natural logarithm with many digits of precision, the Taylor series approach is not efficient since the convergence is slow. Especially if <span class="texhtml mvar" style="font-style:italic;">x</span> is near 1, a good alternative is to use <a href="/wiki/Halley%27s_method" title="Halley's method">Halley's method</a> or <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> to invert the exponential function, because the series of the exponential function converges more quickly. For finding the value of <span class="texhtml mvar" style="font-style:italic;">y</span> to give <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(y)-x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(y)-x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c4eddbf4cde79ab64cb6a18cf25dec3f1d2f74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.948ex; height:2.843ex;" alt="{\displaystyle \exp(y)-x=0}"></span> using Halley's method, or equivalently to give <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(y/2)-x\exp(-y/2)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(y/2)-x\exp(-y/2)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202da8992c9ae8ac73f2e3352c09bca2339e514a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.31ex; height:2.843ex;" alt="{\displaystyle \exp(y/2)-x\exp(-y/2)=0}"></span> using Newton's method, the iteration simplifies to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{n+1}=y_{n}+2\cdot {\frac {x-\exp(y_{n})}{x+\exp(y_{n})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{n+1}=y_{n}+2\cdot {\frac {x-\exp(y_{n})}{x+\exp(y_{n})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64d8c2c100fd228a36d9ac9bfafae8ee0625d62" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.322ex; height:6.509ex;" alt="{\displaystyle y_{n+1}=y_{n}+2\cdot {\frac {x-\exp(y_{n})}{x+\exp(y_{n})}}}"></span> which has <a href="/wiki/Cubic_convergence" class="mw-redirect" title="Cubic convergence">cubic convergence</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df055b8e294310e6785701c1c67105e109191d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \ln(x)}"></span>. </p><p>Another alternative for extremely high precision calculation is the formula<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x\approx {\frac {\pi }{2M(1,4/s)}}-m\ln 2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>M</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>m</mi> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x\approx {\frac {\pi }{2M(1,4/s)}}-m\ln 2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbd58c12155b717e425a232b9d6d45597430449" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.02ex; height:5.509ex;" alt="{\displaystyle \ln x\approx {\frac {\pi }{2M(1,4/s)}}-m\ln 2,}"></span> where <span class="texhtml mvar" style="font-style:italic;">M</span> denotes the <a href="/wiki/Arithmetic-geometric_mean" class="mw-redirect" title="Arithmetic-geometric mean">arithmetic-geometric mean</a> of 1 and <span class="texhtml">4/<i>s</i></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=x2^{m}>2^{p/2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>x</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=x2^{m}>2^{p/2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e97043486e689440687aae21cc96aae4c4aa8c0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.967ex; height:3.176ex;" alt="{\displaystyle s=x2^{m}>2^{p/2},}"></span> with <span class="texhtml mvar" style="font-style:italic;">m</span> chosen so that <span class="texhtml mvar" style="font-style:italic;">p</span> bits of precision is attained. (For most purposes, the value of 8 for <span class="texhtml mvar" style="font-style:italic;">m</span> is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently. (The constants <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1189ffa454489f9a73b3b6aa79f83eb954bea42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.489ex; height:2.176ex;" alt="{\displaystyle \ln 2}"></span> and <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> can be pre-computed to the desired precision using any of several known quickly converging series.) Or, the following formula can be used: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x={\frac {\pi }{M\left(\theta _{2}^{2}(1/x),\theta _{3}^{2}(1/x)\right)}},\quad x\in (1,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msubsup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x={\frac {\pi }{M\left(\theta _{2}^{2}(1/x),\theta _{3}^{2}(1/x)\right)}},\quad x\in (1,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a762c237eacec8dde938fcf6964362f6e70af4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.657ex; height:6.009ex;" alt="{\displaystyle \ln x={\frac {\pi }{M\left(\theta _{2}^{2}(1/x),\theta _{3}^{2}(1/x)\right)}},\quad x\in (1,\infty )}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{2}(x)=\sum _{n\in \mathbb {Z} }x^{(n+1/2)^{2}},\quad \theta _{3}(x)=\sum _{n\in \mathbb {Z} }x^{n^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> </munder> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{2}(x)=\sum _{n\in \mathbb {Z} }x^{(n+1/2)^{2}},\quad \theta _{3}(x)=\sum _{n\in \mathbb {Z} }x^{n^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48135f87a5330a2fd822efda1872654f76337dad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.388ex; height:6.009ex;" alt="{\displaystyle \theta _{2}(x)=\sum _{n\in \mathbb {Z} }x^{(n+1/2)^{2}},\quad \theta _{3}(x)=\sum _{n\in \mathbb {Z} }x^{n^{2}}}"></span> are the <a href="/wiki/Theta_function#Auxiliary_functions" title="Theta function">Jacobi theta functions</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Based on a proposal by <a href="/wiki/William_Kahan" title="William Kahan">William Kahan</a> and first implemented in the <a href="/wiki/Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a> <a href="/wiki/HP-41C" title="HP-41C">HP-41C</a> calculator in 1979 (referred to under "LN1" in the display, only), some calculators, <a href="/wiki/Operating_system" title="Operating system">operating systems</a> (for example <a href="/wiki/Berkeley_UNIX_4.3BSD" class="mw-redirect" title="Berkeley UNIX 4.3BSD">Berkeley UNIX 4.3BSD</a><sup id="cite_ref-Beebe_2017_18-0" class="reference"><a href="#cite_note-Beebe_2017-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>), <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a> and programming languages (for example <a href="/wiki/C99" title="C99">C99</a><sup id="cite_ref-Beebe_2002_19-0" class="reference"><a href="#cite_note-Beebe_2002-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>) provide a special <b>natural logarithm plus 1</b> function, alternatively named <b>LNP1</b>,<sup id="cite_ref-HP48_AUR_20-0" class="reference"><a href="#cite_note-HP48_AUR-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-HP50_AUR_21-0" class="reference"><a href="#cite_note-HP50_AUR-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> or <b>log1p</b><sup id="cite_ref-Beebe_2002_19-1" class="reference"><a href="#cite_note-Beebe_2002-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> to give more accurate results for logarithms close to zero by passing arguments <span class="texhtml mvar" style="font-style:italic;">x</span>, also close to zero, to a function <span class="texhtml">log1p(<i>x</i>)</span>, which returns the value <span class="texhtml">ln(1+<i>x</i>)</span>, instead of passing a value <span class="texhtml mvar" style="font-style:italic;">y</span> close to 1 to a function returning <span class="texhtml">ln(<i>y</i>)</span>.<sup id="cite_ref-Beebe_2002_19-2" class="reference"><a href="#cite_note-Beebe_2002-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-HP48_AUR_20-1" class="reference"><a href="#cite_note-HP48_AUR-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-HP50_AUR_21-1" class="reference"><a href="#cite_note-HP50_AUR-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> The function <span class="texhtml">log1p</span> avoids in the floating point arithmetic a near cancelling of the absolute term 1 with the second term from the Taylor expansion of the natural logarithm. This keeps the argument, the result, and intermediate steps all close to zero where they can be most accurately represented as floating-point numbers.<sup id="cite_ref-HP48_AUR_20-2" class="reference"><a href="#cite_note-HP48_AUR-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-HP50_AUR_21-2" class="reference"><a href="#cite_note-HP50_AUR-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>In addition to base <span class="texhtml mvar" style="font-style:italic;">e</span>, the <a href="/wiki/IEEE_754-2008" class="mw-redirect" title="IEEE 754-2008">IEEE 754-2008</a> standard defines similar logarithmic functions near 1 for <a href="/wiki/Binary_logarithm" title="Binary logarithm">binary</a> and <a href="/wiki/Decimal_logarithm" class="mw-redirect" title="Decimal logarithm">decimal logarithms</a>: <span class="texhtml">log<sub>2</sub>(1 + <i>x</i>)</span> and <span class="texhtml">log<sub>10</sub>(1 + <i>x</i>)</span>. </p><p>Similar inverse functions named "<a href="/wiki/Expm1" class="mw-redirect" title="Expm1">expm1</a>",<sup id="cite_ref-Beebe_2002_19-3" class="reference"><a href="#cite_note-Beebe_2002-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> "expm"<sup id="cite_ref-HP48_AUR_20-3" class="reference"><a href="#cite_note-HP48_AUR-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-HP50_AUR_21-3" class="reference"><a href="#cite_note-HP50_AUR-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> or "exp1m" exist as well, all with the meaning of <span class="texhtml">expm1(<i>x</i>) = exp(<i>x</i>) − 1</span>.<sup id="cite_ref-Alternative_funcs_22-0" class="reference"><a href="#cite_note-Alternative_funcs-22"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </p><p>An identity in terms of the <a href="/wiki/Artanh" class="mw-redirect" title="Artanh">inverse hyperbolic tangent</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {log1p} (x)=\log(1+x)=2~\mathrm {artanh} \left({\frac {x}{2+x}}\right)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">g</mi> <mn>1</mn> <mi mathvariant="normal">p</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">h</mi> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>2</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {log1p} (x)=\log(1+x)=2~\mathrm {artanh} \left({\frac {x}{2+x}}\right)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dee9a12b98b6cc0afd3ac737c4822f40d8d80b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.744ex; height:6.176ex;" alt="{\displaystyle \mathrm {log1p} (x)=\log(1+x)=2~\mathrm {artanh} \left({\frac {x}{2+x}}\right)\,,}"></span> gives a high precision value for small values of <span class="texhtml mvar" style="font-style:italic;">x</span> on systems that do not implement <span class="texhtml">log1p(<i>x</i>)</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Computational_complexity">Computational complexity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=13" title="Edit section: Computational complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Computational_complexity_of_mathematical_operations" title="Computational complexity of mathematical operations">Computational complexity of mathematical operations</a></div> <p>The <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity</a> of computing the natural logarithm using the <a href="/wiki/Arithmetic%E2%80%93geometric_mean" title="Arithmetic–geometric mean">arithmetic-geometric mean</a> (for both of the above methods) is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{O}}{\bigl (}M(n)\ln n{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>O</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mi>ln</mi> <mo>⁡</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{O}}{\bigl (}M(n)\ln n{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a94a7bbfb35d97aad9e5265a4067aadc2b478ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.692ex; height:3.176ex;" alt="{\displaystyle {\text{O}}{\bigl (}M(n)\ln n{\bigr )}}"></span>. Here, <span class="texhtml mvar" style="font-style:italic;">n</span> is the number of digits of precision at which the natural logarithm is to be evaluated, and <span class="texhtml"><i>M</i>(<i>n</i>)</span> is the computational complexity of multiplying two <span class="texhtml mvar" style="font-style:italic;">n</span>-digit numbers. </p> <div class="mw-heading mw-heading2"><h2 id="Continued_fractions">Continued fractions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=14" title="Edit section: Continued fractions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While no simple <a href="/wiki/Continued_fraction" title="Continued fraction">continued fractions</a> are available, several <a href="/wiki/Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">generalized continued fractions</a> exist, including: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln(1+x)&={\frac {x^{1}}{1}}-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+{\frac {x^{5}}{5}}-\cdots \\[5pt]&={\cfrac {x}{1-0x+{\cfrac {1^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+{\cfrac {4^{2}x}{5-4x+\ddots }}}}}}}}}}\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="0.8em 0.3em" 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> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mn>0</mn> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln(1+x)&={\frac {x^{1}}{1}}-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+{\frac {x^{5}}{5}}-\cdots \\[5pt]&={\cfrac {x}{1-0x+{\cfrac {1^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+{\cfrac {4^{2}x}{5-4x+\ddots }}}}}}}}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92f9f9bda019d60b5ac5d5fd29ea2dd952c5b90a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.505ex; width:66.767ex; height:30.176ex;" alt="{\displaystyle {\begin{aligned}\ln(1+x)&={\frac {x^{1}}{1}}-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+{\frac {x^{5}}{5}}-\cdots \\[5pt]&={\cfrac {x}{1-0x+{\cfrac {1^{2}x}{2-1x+{\cfrac {2^{2}x}{3-2x+{\cfrac {3^{2}x}{4-3x+{\cfrac {4^{2}x}{5-4x+\ddots }}}}}}}}}}\end{aligned}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln \left(1+{\frac {x}{y}}\right)&={\cfrac {x}{y+{\cfrac {1x}{2+{\cfrac {1x}{3y+{\cfrac {2x}{2+{\cfrac {2x}{5y+{\cfrac {3x}{2+\ddots }}}}}}}}}}}}\\[5pt]&={\cfrac {2x}{2y+x-{\cfrac {(1x)^{2}}{3(2y+x)-{\cfrac {(2x)^{2}}{5(2y+x)-{\cfrac {(3x)^{2}}{7(2y+x)-\ddots }}}}}}}}\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="0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mi>y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>x</mi> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln \left(1+{\frac {x}{y}}\right)&={\cfrac {x}{y+{\cfrac {1x}{2+{\cfrac {1x}{3y+{\cfrac {2x}{2+{\cfrac {2x}{5y+{\cfrac {3x}{2+\ddots }}}}}}}}}}}}\\[5pt]&={\cfrac {2x}{2y+x-{\cfrac {(1x)^{2}}{3(2y+x)-{\cfrac {(2x)^{2}}{5(2y+x)-{\cfrac {(3x)^{2}}{7(2y+x)-\ddots }}}}}}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90abfa2132828fc8eea5d3551dfa4df25dbdfa87" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -23.171ex; width:67.931ex; height:47.509ex;" alt="{\displaystyle {\begin{aligned}\ln \left(1+{\frac {x}{y}}\right)&={\cfrac {x}{y+{\cfrac {1x}{2+{\cfrac {1x}{3y+{\cfrac {2x}{2+{\cfrac {2x}{5y+{\cfrac {3x}{2+\ddots }}}}}}}}}}}}\\[5pt]&={\cfrac {2x}{2y+x-{\cfrac {(1x)^{2}}{3(2y+x)-{\cfrac {(2x)^{2}}{5(2y+x)-{\cfrac {(3x)^{2}}{7(2y+x)-\ddots }}}}}}}}\end{aligned}}}"></span> </p><p>These continued fractions—particularly the last—converge rapidly for values close to 1. However, the natural logarithms of much larger numbers can easily be computed, by repeatedly adding those of smaller numbers, with similarly rapid convergence. </p><p>For example, since 2 = 1.25<sup>3</sup> × 1.024, the <a href="/wiki/Natural_logarithm_of_2" title="Natural logarithm of 2">natural logarithm of 2</a> can be computed as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln 2&=3\ln \left(1+{\frac {1}{4}}\right)+\ln \left(1+{\frac {3}{125}}\right)\\[8pt]&={\cfrac {6}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {6}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\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="1.1em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>125</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>63</mn> <mo>−</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>253</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>759</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1265</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1771</mn> <mo>−</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln 2&=3\ln \left(1+{\frac {1}{4}}\right)+\ln \left(1+{\frac {3}{125}}\right)\\[8pt]&={\cfrac {6}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {6}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc10de9595aca079ef56e7b76a2a23af56e453da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.988ex; margin-bottom: -0.183ex; width:70.607ex; height:27.509ex;" alt="{\displaystyle {\begin{aligned}\ln 2&=3\ln \left(1+{\frac {1}{4}}\right)+\ln \left(1+{\frac {3}{125}}\right)\\[8pt]&={\cfrac {6}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {6}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\end{aligned}}}"></span> </p><p>Furthermore, since 10 = 1.25<sup>10</sup> × 1.024<sup>3</sup>, even the natural logarithm of 10 can be computed similarly as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\ln 10&=10\ln \left(1+{\frac {1}{4}}\right)+3\ln \left(1+{\frac {3}{125}}\right)\\[10pt]&={\cfrac {20}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {18}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\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="1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>ln</mi> <mo>⁡</mo> <mn>10</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>10</mn> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>3</mn> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>125</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>45</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>63</mn> <mo>−</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>253</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>759</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1265</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1771</mn> <mo>−</mo> <mo>⋱</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ln 10&=10\ln \left(1+{\frac {1}{4}}\right)+3\ln \left(1+{\frac {3}{125}}\right)\\[10pt]&={\cfrac {20}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {18}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/931b5e1a786450547bd77e466677d9a983974886" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.22ex; margin-bottom: -0.285ex; width:71.77ex; height:28.176ex;" alt="{\displaystyle {\begin{aligned}\ln 10&=10\ln \left(1+{\frac {1}{4}}\right)+3\ln \left(1+{\frac {3}{125}}\right)\\[10pt]&={\cfrac {20}{9-{\cfrac {1^{2}}{27-{\cfrac {2^{2}}{45-{\cfrac {3^{2}}{63-\ddots }}}}}}}}+{\cfrac {18}{253-{\cfrac {3^{2}}{759-{\cfrac {6^{2}}{1265-{\cfrac {9^{2}}{1771-\ddots }}}}}}}}.\end{aligned}}}"></span> The reciprocal of the natural logarithm can be also written in this way: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\ln(x)}}={\frac {2x}{x^{2}-1}}{\sqrt {{\frac {1}{2}}+{\frac {x^{2}+1}{4x}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {x^{2}+1}{4x}}}}}}\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>x</mi> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>x</mi> </mrow> </mfrac> </mrow> </msqrt> </mrow> </msqrt> </mrow> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\ln(x)}}={\frac {2x}{x^{2}-1}}{\sqrt {{\frac {1}{2}}+{\frac {x^{2}+1}{4x}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {x^{2}+1}{4x}}}}}}\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9059b4316dee7218119cb2e6bbf3bdba8a2343ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.279ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{\ln(x)}}={\frac {2x}{x^{2}-1}}{\sqrt {{\frac {1}{2}}+{\frac {x^{2}+1}{4x}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {x^{2}+1}{4x}}}}}}\ldots }"></span> </p><p>For example: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\ln(2)}}={\frac {4}{3}}{\sqrt {{\frac {1}{2}}+{\frac {5}{8}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {5}{8}}}}}}\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <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>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> </mrow> </msqrt> </mrow> </msqrt> </mrow> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\ln(2)}}={\frac {4}{3}}{\sqrt {{\frac {1}{2}}+{\frac {5}{8}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {5}{8}}}}}}\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6e6c401a41778cb66e456ad990d5c1081b57e9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.439ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{\ln(2)}}={\frac {4}{3}}{\sqrt {{\frac {1}{2}}+{\frac {5}{8}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {5}{8}}}}}}\ldots }"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Complex_logarithms">Complex logarithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=15" title="Edit section: Complex logarithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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> <p>The exponential function can be extended to a function which gives a <a href="/wiki/Complex_number" title="Complex number">complex number</a> as <span class="texhtml"><i>e</i><sup><i>z</i></sup></span> for any arbitrary complex number <span class="texhtml mvar" style="font-style:italic;">z</span>; simply use the infinite series with <span class="texhtml mvar" style="font-style:italic;">x</span>=z complex. This exponential function can be inverted to form a complex logarithm that exhibits most of the properties of the ordinary logarithm. There are two difficulties involved: no <span class="texhtml mvar" style="font-style:italic;">x</span> has <span class="texhtml"><i>e</i><sup><i>x</i></sup> = 0</span>; and it turns out that <span class="texhtml"><i>e</i><sup>2<i>iπ</i></sup> = 1 = <i>e</i><sup>0</sup></span>. Since the multiplicative property still works for the complex exponential function, <span class="texhtml"><i>e</i><sup><i>z</i></sup> = <i>e</i><sup><i>z</i>+2<i>kiπ</i></sup></span>, for all complex <span class="texhtml mvar" style="font-style:italic;">z</span> and integers <span class="texhtml mvar" style="font-style:italic;">k</span>. </p><p>So the logarithm cannot be defined for the whole <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, and even then it is <a href="/wiki/Multi-valued" class="mw-redirect" title="Multi-valued">multi-valued</a>—any complex logarithm can be changed into an "equivalent" logarithm by adding any integer multiple of <span class="texhtml">2<i>iπ</i></span> at will. The complex logarithm can only be single-valued on the <a href="/wiki/Complex_plane#Cutting_the_plane" title="Complex plane">cut plane</a>. For example, <span class="texhtml">ln <i>i</i> = <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><span class="sfrac">⁠<span class="tion"><span class="num"><i>iπ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> or <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5<i>iπ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> or <span class="texhtml">-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3<i>iπ</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, etc.; and although <span class="texhtml"><i>i</i><sup>4</sup> = 1, 4 ln <i>i</i></span> can be defined as <span class="texhtml">2<i>iπ</i></span>, or <span class="texhtml">10<i>iπ</i></span> or <span class="texhtml">−6<i>iπ</i></span>, and so on. </p> <ul class="gallery mw-gallery-packed"> <li class="gallerycaption">Plots of the natural logarithm function on the complex plane (<a href="/wiki/Principal_branch" title="Principal branch">principal branch</a>)</li> <li class="gallerybox" style="width: 174.66666666667px"> <div class="thumb" style="width: 172.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:NaturalLogarithmRe.png" class="mw-file-description" title="z = Re(ln(x + yi))"><img alt="z = Re(ln(x + yi))" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/NaturalLogarithmRe.png/259px-NaturalLogarithmRe.png" decoding="async" width="173" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/NaturalLogarithmRe.png/388px-NaturalLogarithmRe.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/NaturalLogarithmRe.png/518px-NaturalLogarithmRe.png 2x" data-file-width="1249" data-file-height="869" /></a></span></div> <div class="gallerytext"><span class="texhtml"><i>z</i> = Re(ln(<i>x</i> + <i>yi</i>))</span> </div> </li> <li class="gallerybox" style="width: 148px"> <div class="thumb" style="width: 146px;"><span typeof="mw:File"><a href="/wiki/File:NaturalLogarithmImAbs.png" class="mw-file-description" title="z = |(Im(ln(x + yi)))|"><img alt="z = |(Im(ln(x + yi)))|" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/NaturalLogarithmImAbs.png/219px-NaturalLogarithmImAbs.png" decoding="async" width="146" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/NaturalLogarithmImAbs.png/328px-NaturalLogarithmImAbs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/NaturalLogarithmImAbs.png/437px-NaturalLogarithmImAbs.png 2x" data-file-width="1180" data-file-height="972" /></a></span></div> <div class="gallerytext"><span class="texhtml"><i>z</i> = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">(Im(ln(<i>x</i> + <i>yi</i>)))</span>|</span> </div> </li> <li class="gallerybox" style="width: 148.66666666667px"> <div class="thumb" style="width: 146.66666666667px;"><span typeof="mw:File"><a href="/wiki/File:NaturalLogarithmAbs.png" class="mw-file-description" title="z = |(ln(x + yi))|"><img alt="z = |(ln(x + yi))|" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/NaturalLogarithmAbs.png/220px-NaturalLogarithmAbs.png" decoding="async" width="147" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/NaturalLogarithmAbs.png/330px-NaturalLogarithmAbs.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/NaturalLogarithmAbs.png/440px-NaturalLogarithmAbs.png 2x" data-file-width="1250" data-file-height="1023" /></a></span></div> <div class="gallerytext"><span class="texhtml"><i>z</i> = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">(ln(<i>x</i> + <i>yi</i>))</span>|</span></div> </li> <li class="gallerybox" style="width: 177.33333333333px"> <div class="thumb" style="width: 175.33333333333px;"><span typeof="mw:File"><a href="/wiki/File:NaturalLogarithmAll.png" class="mw-file-description" title="Superposition of the previous three graphs"><img alt="Superposition of the previous three graphs" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/NaturalLogarithmAll.png/263px-NaturalLogarithmAll.png" decoding="async" width="176" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/NaturalLogarithmAll.png/394px-NaturalLogarithmAll.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/NaturalLogarithmAll.png/525px-NaturalLogarithmAll.png 2x" data-file-width="1244" data-file-height="853" /></a></span></div> <div class="gallerytext"> Superposition of the previous three graphs</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Iterated_logarithm" title="Iterated logarithm">Iterated logarithm</a></li> <li><a href="/wiki/Napierian_logarithm" title="Napierian logarithm">Napierian logarithm</a></li> <li><a href="/wiki/List_of_logarithmic_identities" title="List of logarithmic identities">List of logarithmic identities</a></li> <li><a href="/wiki/Logarithm_of_a_matrix" title="Logarithm of a matrix">Logarithm of a matrix</a></li> <li><a href="/wiki/Exponential_map_(Lie_theory)#Logarithmic_coordinates" title="Exponential map (Lie theory)">Logarithmic coordinates</a> of an element of a Lie group.</li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li> <li><a href="/wiki/Logarithmic_integral_function" title="Logarithmic integral function">Logarithmic integral function</a></li> <li><a href="/wiki/Nicholas_Mercator" title="Nicholas Mercator">Nicholas Mercator</a> – first to use the term natural logarithm</li> <li><a href="/wiki/Polylogarithm" title="Polylogarithm">Polylogarithm</a></li> <li><a href="/wiki/Von_Mangoldt_function" title="Von Mangoldt function">Von Mangoldt function</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=17" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"> Including <a href="/wiki/C_(programming_language)" title="C (programming language)">C</a>, <a href="/wiki/C%2B%2B" title="C++">C++</a>, <a href="/wiki/SAS_System" class="mw-redirect" title="SAS System">SAS</a>, <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a>, <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>, <a href="/wiki/Fortran" title="Fortran">Fortran</a>, and some <a href="/wiki/BASIC_programming_language" class="mw-redirect" title="BASIC programming language">BASIC</a> dialects</span> </li> <li id="cite_note-Alternative_funcs-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Alternative_funcs_22-0">^</a></b></span> <span class="reference-text">For a similar approach to reduce <a href="/wiki/Round-off_error" title="Round-off error">round-off errors</a> of calculations for certain input values see <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric functions</a> like <a href="/wiki/Versine" title="Versine">versine</a>, <a href="/wiki/Vercosine" class="mw-redirect" title="Vercosine">vercosine</a>, <a href="/wiki/Coversine" class="mw-redirect" title="Coversine">coversine</a>, <a href="/wiki/Covercosine" class="mw-redirect" title="Covercosine">covercosine</a>, <a href="/wiki/Haversine" class="mw-redirect" title="Haversine">haversine</a>, <a href="/wiki/Havercosine" class="mw-redirect" title="Havercosine">havercosine</a>, <a href="/wiki/Hacoversine" class="mw-redirect" title="Hacoversine">hacoversine</a>, <a href="/wiki/Hacovercosine" class="mw-redirect" title="Hacovercosine">hacovercosine</a>, <a href="/wiki/Exsecant" title="Exsecant">exsecant</a> and <a href="/wiki/Excosecant" class="mw-redirect" title="Excosecant">excosecant</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Natural_logarithm&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFSloane_"A001113"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A001113">"Sequence A001113 (Decimal expansion of e)"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA001113%26%23x20%3B%28Decimal+expansion+of+e%29&rft_id=https%3A%2F%2Foeis.org%2FA001113&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford 1975, footnote to paragraph 1.7: "<i>log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest</i>".</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMortimer2005" class="citation book cs1">Mortimer, Robert G. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGoSv5tmATsC"><i>Mathematics for physical chemistry</i></a> (3rd ed.). <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. p. 9. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-508347-5" title="Special:BookSources/0-12-508347-5"><bdi>0-12-508347-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+physical+chemistry&rft.pages=9&rft.edition=3rd&rft.pub=Academic+Press&rft.date=2005&rft.isbn=0-12-508347-5&rft.aulast=Mortimer&rft.aufirst=Robert+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnGoSv5tmATsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGoSv5tmATsC&pg=PA9">Extract of page 9</a></span> </li> <li id="cite_note-:1-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:1_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/NaturalLogarithm.html">"Natural Logarithm"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Natural+Logarithm&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FNaturalLogarithm.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-:2-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/logarithm">"Rules, Examples, & Formulas"</a>. Logarithm. <i>Encyclopedia Britannica</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+Britannica&rft.atitle=Rules%2C+Examples%2C+%26+Formulas&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Flogarithm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurn2001" class="citation journal cs1">Burn, R.P. (2001). "Alphonse Antonio de Sarasa and logarithms". <i><a href="/wiki/Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a></i>. <b>28</b>: <span class="nowrap">1–</span>17. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fhmat.2000.2295">10.1006/hmat.2000.2295</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=Alphonse+Antonio+de+Sarasa+and+logarithms&rft.volume=28&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E17&rft.date=2001&rft_id=info%3Adoi%2F10.1006%2Fhmat.2000.2295&rft.aulast=Burn&rft.aufirst=R.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson2001" class="citation web cs1">O'Connor, J. J.; Robertson, E. F. (September 2001). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/HistTopics/e.html">"The number e"</a>. The MacTutor History of Mathematics archive<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-02-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+number+e&rft.pub=The+MacTutor+History+of+Mathematics+archive&rft.date=2001-09&rft.aulast=O%27Connor&rft.aufirst=J.+J.&rft.au=Robertson%2C+E.+F.&rft_id=http%3A%2F%2Fwww-history.mcs.st-and.ac.uk%2FHistTopics%2Fe.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-Cajori-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cajori_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cajori_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1991" class="citation book cs1"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mGJRjIC9fZgC"><i>A History of Mathematics</i></a> (5th ed.). AMS Bookstore. p. 152. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-2102-4" title="Special:BookSources/0-8218-2102-4"><bdi>0-8218-2102-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=152&rft.edition=5th&rft.pub=AMS+Bookstore&rft.date=1991&rft.isbn=0-8218-2102-4&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmGJRjIC9fZgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math2.org/math/expansion/log.htm">"<span class="cs1-kern-left"></span>"Logarithmic Expansions" at Math2.org"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=%22Logarithmic+Expansions%22+at+Math2.org&rft_id=http%3A%2F%2Fwww.math2.org%2Fmath%2Fexpansion%2Flog.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, Introductio in Analysin Infinitorum. Tomus Primus. Bousquet, Lausanne 1748. Exemplum 1, p. 228; quoque in: Opera Omnia, Series Prima, Opera Mathematica, Volumen Octavum, Teubner 1922</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRUFFA" class="citation web cs1">RUFFA, Anthony. <a rel="nofollow" class="external text" href="https://www.emis.de/journals/HOA/IJMMS/2004/65-683653.pdf">"A PROCEDURE FOR GENERATING INFINITE SERIES IDENTITIES"</a> <span class="cs1-format">(PDF)</span>. <i>International Journal of Mathematics and Mathematical Sciences</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-02-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=International+Journal+of+Mathematics+and+Mathematical+Sciences&rft.atitle=A+PROCEDURE+FOR+GENERATING+INFINITE+SERIES+IDENTITIES&rft.aulast=RUFFA&rft.aufirst=Anthony&rft_id=https%3A%2F%2Fwww.emis.de%2Fjournals%2FHOA%2FIJMMS%2F2004%2F65-683653.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span> (Page 3654, equation 2.6)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">For a detailed proof see for instance: George B. Thomas, Jr and Ross L. Finney, <i>Calculus and Analytic Geometry</i>, 5th edition, Addison-Wesley 1979, Section 6-5 pages 305-306.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSloane_"A002392"" class="citation web cs1"><a href="/wiki/Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A002392">"Sequence A002392 (Decimal expansion of natural logarithm of 10)"</a>. <i>The <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+On-Line+Encyclopedia+of+Integer+Sequences&rft.atitle=Sequence%26%23x20%3BA002392%26%23x20%3B%28Decimal+expansion+of+natural+logarithm+of+10%29&rft_id=https%3A%2F%2Foeis.org%2FA002392&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSasakiKanada1982" class="citation journal cs1">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>. <i>Journal of Information Processing</i>. <b>5</b> (4): <span class="nowrap">247–</span>250<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-03-30</span></span>.</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=%3Cspan+class%3D%22nowrap%22%3E247-%3C%2Fspan%3E250&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%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAhrendt1999" class="citation book cs1">Ahrendt, Timm (1999). "Fast Computations of the Exponential Function". <i>Stacs 99</i>. Lecture Notes in Computer Science. Vol. 1564. pp. <span class="nowrap">302–</span>312. <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.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E302-%3C%2Fspan%3E312&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%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorweinBorwein1987" class="citation book cs1">Borwein, Jonathan M.; Borwein, Peter B. (1987). <i>Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity</i> (First ed.). Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-83138-7" title="Special:BookSources/0-471-83138-7"><bdi>0-471-83138-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+and+the+AGM%3A+A+Study+in+Analytic+Number+Theory+and+Computational+Complexity&rft.edition=First&rft.pub=Wiley-Interscience&rft.date=1987&rft.isbn=0-471-83138-7&rft.aulast=Borwein&rft.aufirst=Jonathan+M.&rft.au=Borwein%2C+Peter+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span> page 225</span> </li> <li id="cite_note-Beebe_2017-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-Beebe_2017_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeebe2017" class="citation book cs1">Beebe, Nelson H. F. (2017-08-22). "Chapter 10.4. Logarithm near one". <i>The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library</i> (1 ed.). Salt Lake City, UT, USA: <a href="/wiki/Springer_International_Publishing_AG" class="mw-redirect" title="Springer International Publishing AG">Springer International Publishing AG</a>. pp. <span class="nowrap">290–</span>292. <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%2F978-3-319-64110-2">10.1007/978-3-319-64110-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-64109-6" title="Special:BookSources/978-3-319-64109-6"><bdi>978-3-319-64109-6</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2017947446">2017947446</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:30244721">30244721</a>. <q>In 1987, Berkeley UNIX 4.3BSD introduced the log1p() function</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+10.4.+Logarithm+near+one&rft.btitle=The+Mathematical-Function+Computation+Handbook+-+Programming+Using+the+MathCW+Portable+Software+Library&rft.place=Salt+Lake+City%2C+UT%2C+USA&rft.pages=%3Cspan+class%3D%22nowrap%22%3E290-%3C%2Fspan%3E292&rft.edition=1&rft.pub=Springer+International+Publishing+AG&rft.date=2017-08-22&rft_id=info%3Adoi%2F10.1007%2F978-3-319-64110-2&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A30244721%23id-name%3DS2CID&rft_id=info%3Alccn%2F2017947446&rft.isbn=978-3-319-64109-6&rft.aulast=Beebe&rft.aufirst=Nelson+H.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-Beebe_2002-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-Beebe_2002_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Beebe_2002_19-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Beebe_2002_19-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Beebe_2002_19-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeebe2002" class="citation web cs1">Beebe, Nelson H. F. (2002-07-09). <a rel="nofollow" class="external text" href="http://www.math.utah.edu/~beebe/reports/expm1.pdf">"Computation of expm1 = exp(x)−1"</a> <span class="cs1-format">(PDF)</span>. 1.00. Salt Lake City, Utah, USA: Department of Mathematics, Center for Scientific Computing, <a href="/wiki/University_of_Utah" title="University of Utah">University of Utah</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2015-11-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Computation+of+expm1+%3D+exp%28x%29%E2%88%921&rft.place=Salt+Lake+City%2C+Utah%2C+USA&rft.series=1.00&rft.pub=Department+of+Mathematics%2C+Center+for+Scientific+Computing%2C+University+of+Utah&rft.date=2002-07-09&rft.aulast=Beebe&rft.aufirst=Nelson+H.+F.&rft_id=http%3A%2F%2Fwww.math.utah.edu%2F~beebe%2Freports%2Fexpm1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-HP48_AUR-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-HP48_AUR_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HP48_AUR_20-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-HP48_AUR_20-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-HP48_AUR_20-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://www.hpcalc.org/details.php?id=6036"><i>HP 48G Series – Advanced User's Reference Manual (AUR)</i></a> (4 ed.). <a href="/wiki/Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a>. December 1994 [1993]. HP 00048-90136, 0-88698-01574-2<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-09-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=HP+48G+Series+%E2%80%93+Advanced+User%27s+Reference+Manual+%28AUR%29&rft.edition=4&rft.pub=Hewlett-Packard&rft.date=1994-12&rft_id=http%3A%2F%2Fwww.hpcalc.org%2Fdetails.php%3Fid%3D6036&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span></span> </li> <li id="cite_note-HP50_AUR-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-HP50_AUR_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HP50_AUR_21-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-HP50_AUR_21-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-HP50_AUR_21-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://www.hpcalc.org/details.php?id=7141"><i>HP 50g / 49g+ / 48gII graphing calculator advanced user's reference manual (AUR)</i></a> (2 ed.). <a href="/wiki/Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a>. 2009-07-14 [2005]. HP F2228-90010<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-10-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=HP+50g+%2F+49g%2B+%2F+48gII+graphing+calculator+advanced+user%27s+reference+manual+%28AUR%29&rft.edition=2&rft.pub=Hewlett-Packard&rft.date=2009-07-14&rft_id=http%3A%2F%2Fwww.hpcalc.org%2Fdetails.php%3Fid%3D7141&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANatural+logarithm" class="Z3988"></span> <a rel="nofollow" class="external text" href="http://holyjoe.net/hp/HP_50g_AUR_v2_English_searchable.pdf">Searchable PDF</a></span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox 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class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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:Calculus_topics" title="Template:Calculus topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus_topics" title="Template talk:Calculus topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus_topics" title="Special:EditPage/Template:Calculus topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Calculus249" style="font-size:114%;margin:0 4em"><a href="/wiki/Calculus" title="Calculus">Calculus</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Binomial_theorem" title="Binomial theorem">Binomial theorem</a></li> <li><a href="/wiki/Concave_function" title="Concave function">Concave function</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous function</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Finite_difference" title="Finite difference">Finite difference</a></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables and bound variables</a></li> <li><a href="/wiki/Graph_of_a_function" title="Graph of a function">Graph of a function</a></li> <li><a href="/wiki/Linear_function" title="Linear function">Linear function</a></li> <li><a href="/wiki/Radian" title="Radian">Radian</a></li> <li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Secant_line" title="Secant line">Secant</a></li> <li><a href="/wiki/Slope" title="Slope">Slope</a></li> <li><a href="/wiki/Tangent" title="Tangent">Tangent</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">Limits</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a></li> <li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limit of a function</a> <ul><li><a href="/wiki/One-sided_limit" title="One-sided limit">One-sided limit</a></li></ul></li> <li><a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Limit of a sequence</a></li> <li><a href="/wiki/Order_of_approximation" title="Order of approximation">Order of approximation</a></li> <li><a href="/wiki/(%CE%B5,_%CE%B4)-definition_of_limit" class="mw-redirect" title="(ε, δ)-definition of limit">(ε, δ)-definition of limit</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</a> <ul><li><a href="/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a></li> <li><a href="/wiki/Newton%27s_notation_for_differentiation" class="mw-redirect" title="Newton's notation for differentiation">Newton's notation</a></li></ul></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules of differentiation</a> <ul><li><a href="/wiki/Linearity_of_differentiation" title="Linearity of differentiation">linearity</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a> <ul><li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz's rule</a></li></ul></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li></ul></li> <li>Other techniques <ul><li><a href="/wiki/Implicit_differentiation" class="mw-redirect" title="Implicit differentiation">Implicit differentiation</a></li> <li><a href="/wiki/Inverse_functions_and_differentiation" class="mw-redirect" title="Inverse functions and differentiation">Inverse functions and differentiation</a></li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li></ul></li> <li><a href="/wiki/Stationary_point" title="Stationary point">Stationary points</a> <ul><li><a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">First derivative test</a></li> <li><a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a></li> <li><a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">Extreme value theorem</a></li> <li><a href="/wiki/Maximum_and_minimum" title="Maximum and minimum">Maximum and minimum</a></li></ul></li> <li>Further applications <ul><li><a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equation</a> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equation</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial differential equation</a></li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">Integral calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Arc_length" title="Arc length">Arc length</a></li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Integral#Properties" title="Integral">Basic properties</a></li> <li><a href="/wiki/Constant_of_integration" title="Constant of integration">Constant of integration</a></li> <li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a> <ul><li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Differentiating under the integral sign</a></li></ul></li> <li><a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Integration by substitution</a> <ul><li><a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a></li> <li><a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a></li> <li><a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br />and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a class="mw-selflink selflink">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a href="/wiki/Regiomontanus%27_angle_maximization_problem" title="Regiomontanus' angle maximization problem">Regiomontanus' angle maximization problem</a></li> <li><a href="/wiki/Steinmetz_solid" title="Steinmetz solid">Steinmetz solid</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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Natural logarithm - Wikipedia