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terminivõistlus\u003C/a\u003E\u003C/b\u003E!\u003C/small\u003E\u003C/span\u003E\n\u003C/p\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sait"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Sisukord" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Sisukord</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">vii külgpaanile</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">peida</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Algus</div> </a> </li> <li id="toc-Logaritmi_omadused" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Logaritmi_omadused"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Logaritmi omadused</span> </div> </a> <button aria-controls="toc-Logaritmi_omadused-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>Lülita ümber alaosa "Logaritmi omadused"</span> </button> <ul id="toc-Logaritmi_omadused-sublist" class="vector-toc-list"> <li id="toc-Logaritm_korrutisest_ja_jagatisest" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logaritm_korrutisest_ja_jagatisest"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Logaritm korrutisest ja jagatisest</span> </div> </a> <ul id="toc-Logaritm_korrutisest_ja_jagatisest-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logaritm_astmest" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logaritm_astmest"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Logaritm astmest</span> </div> </a> <ul id="toc-Logaritm_astmest-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logaritmi_aluse_vahetamine" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logaritmi_aluse_vahetamine"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Logaritmi aluse vahetamine</span> </div> </a> <ul id="toc-Logaritmi_aluse_vahetamine-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Logaritmi_rakendused" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Logaritmi_rakendused"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Logaritmi rakendused</span> </div> </a> <ul id="toc-Logaritmi_rakendused-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vaata_ka" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vaata_ka"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Vaata ka</span> </div> </a> <ul id="toc-Vaata_ka-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Välislingid" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Välislingid"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Välislingid</span> </div> </a> <ul id="toc-Välislingid-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="Sisukord" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Lülita sisukord ümber" > <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">Lülita sisukord ümber</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">Logaritm</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="Mine teises keeles artiklisse. Saadaval 109 keeles" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-109" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">109 keelt</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/Logaritme" title="Logaritme – afrikaani" lang="af" hreflang="af" data-title="Logaritme" data-language-autonym="Afrikaans" data-language-local-name="afrikaani" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Logarithmus" title="Logarithmus – šveitsisaksa" lang="gsw" hreflang="gsw" data-title="Logarithmus" data-language-autonym="Alemannisch" data-language-local-name="šveitsisaksa" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%8E%E1%8C%8B%E1%88%AA%E1%8B%9D%E1%88%9D" title="ሎጋሪዝም – amhara" lang="am" hreflang="am" data-title="ሎጋሪዝም" data-language-autonym="አማርኛ" data-language-local-name="amhara" class="interlanguage-link-target"><span>አማርኛ</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" title="لوغاريتم – araabia" lang="ar" hreflang="ar" data-title="لوغاريتم" data-language-autonym="العربية" data-language-local-name="araabia" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Logaritmo" title="Logaritmo – aragoni" lang="an" hreflang="an" data-title="Logaritmo" data-language-autonym="Aragonés" data-language-local-name="aragoni" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%98%E0%A6%BE%E0%A6%A4%E0%A6%BE%E0%A6%82%E0%A6%95" title="ঘাতাংক – assami" lang="as" hreflang="as" data-title="ঘাতাংক" data-language-autonym="অসমীয়া" data-language-local-name="assami" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Logaritmu" title="Logaritmu – astuuria" lang="ast" hreflang="ast" data-title="Logaritmu" data-language-autonym="Asturianu" data-language-local-name="astuuria" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Loqarifm" title="Loqarifm – aserbaidžaani" lang="az" hreflang="az" data-title="Loqarifm" data-language-autonym="Azərbaycanca" data-language-local-name="aserbaidžaani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logaritma" title="Logaritma – indoneesia" lang="id" hreflang="id" data-title="Logaritma" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneesia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logaritma" title="Logaritma – malai" lang="ms" hreflang="ms" data-title="Logaritma" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B2%E0%A6%97%E0%A6%BE%E0%A6%B0%E0%A6%BF%E0%A6%A6%E0%A6%AE" title="লগারিদম – bengali" lang="bn" hreflang="bn" data-title="লগারিদম" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Logaritma" title="Logaritma – bandžari" lang="bjn" hreflang="bjn" data-title="Logaritma" data-language-autonym="Banjar" data-language-local-name="bandžari" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/T%C3%B9i-s%C3%B2%CD%98" title="Tùi-sò͘ – lõunamini" lang="nan" hreflang="nan" data-title="Tùi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="lõunamini" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – baškiiri" lang="ba" hreflang="ba" data-title="Логарифм" data-language-autonym="Башҡортса" data-language-local-name="baškiiri" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D0%B0%D1%80%D1%8B%D1%84%D0%BC" title="Лагарыфм – valgevene" lang="be" hreflang="be" data-title="Лагарыфм" data-language-autonym="Беларуская" data-language-local-name="valgevene" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%8F%D0%B3%D0%B0%D1%80%D1%8B%D1%82%D0%BC" title="Лягарытм – valgevene (taraškievitsa)" lang="be-tarask" hreflang="be-tarask" data-title="Лягарытм" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="valgevene (taraškievitsa)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – keskbikoli" lang="bcl" hreflang="bcl" data-title="Logaritmo" data-language-autonym="Bikol Central" data-language-local-name="keskbikoli" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Logaritam" title="Logaritam – bosnia" lang="bs" hreflang="bs" data-title="Logaritam" data-language-autonym="Bosanski" data-language-local-name="bosnia" 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" title="Logaritm – bretooni" lang="br" hreflang="br" data-title="Logaritm" data-language-autonym="Brezhoneg" data-language-local-name="bretooni" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D1%8A%D0%BC" title="Логаритъм – bulgaaria" lang="bg" hreflang="bg" data-title="Логаритъм" data-language-autonym="Български" data-language-local-name="bulgaaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – burjaadi" lang="bxr" hreflang="bxr" data-title="Логарифм" data-language-autonym="Буряад" data-language-local-name="burjaadi" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Logaritme" title="Logaritme – katalaani" lang="ca" hreflang="ca" data-title="Logaritme" data-language-autonym="Català" data-language-local-name="katalaani" 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%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – tšuvaši" lang="cv" hreflang="cv" data-title="Логарифм" data-language-autonym="Чӑвашла" data-language-local-name="tšuvaši" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Logaritmus" title="Logaritmus – tšehhi" lang="cs" hreflang="cs" data-title="Logaritmus" data-language-autonym="Čeština" data-language-local-name="tšehhi" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Daraunene" title="Daraunene – šona" lang="sn" hreflang="sn" data-title="Daraunene" data-language-autonym="ChiShona" data-language-local-name="šona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Logarithm" title="Logarithm – kõmri" lang="cy" hreflang="cy" data-title="Logarithm" data-language-autonym="Cymraeg" data-language-local-name="kõmri" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Logaritme" title="Logaritme – taani" lang="da" hreflang="da" data-title="Logaritme" data-language-autonym="Dansk" data-language-local-name="taani" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%84%D9%88%DA%AD%D8%A7%D8%B1%D9%8A%D8%AA%D9%85" title="لوڭاريتم – Maroko araabia" lang="ary" hreflang="ary" data-title="لوڭاريتم" data-language-autonym="الدارجة" data-language-local-name="Maroko araabia" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Logarithmus" title="Logarithmus – saksa" lang="de" hreflang="de" data-title="Logarithmus" data-language-autonym="Deutsch" data-language-local-name="saksa" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%AC%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82" title="Λογάριθμος – kreeka" lang="el" hreflang="el" data-title="Λογάριθμος" data-language-autonym="Ελληνικά" data-language-local-name="kreeka" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Logar%C3%ACtem" title="Logarìtem – emiilia-romanja" lang="egl" hreflang="egl" data-title="Logarìtem" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="emiilia-romanja" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437796 badge-featuredarticle mw-list-item" title="eeskujulik artikkel"><a href="https://en.wikipedia.org/wiki/Logarithm" title="Logarithm – inglise" lang="en" hreflang="en" data-title="Logarithm" data-language-autonym="English" data-language-local-name="inglise" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Logaritmo" title="Logaritmo – hispaania" lang="es" hreflang="es" data-title="Logaritmo" data-language-autonym="Español" data-language-local-name="hispaania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Logaritmo" title="Logaritmo – esperanto" lang="eo" hreflang="eo" data-title="Logaritmo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Logaritmu" title="Logaritmu – estremenju" lang="ext" hreflang="ext" data-title="Logaritmu" data-language-autonym="Estremeñu" data-language-local-name="estremenju" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Logaritmo" title="Logaritmo – baski" lang="eu" hreflang="eu" data-title="Logaritmo" data-language-autonym="Euskara" data-language-local-name="baski" 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" title="لگاریتم – pärsia" lang="fa" hreflang="fa" data-title="لگاریتم" data-language-autonym="فارسی" data-language-local-name="pärsia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Logarithm" title="Logarithm – Fidži hindi" lang="hif" hreflang="hif" data-title="Logarithm" data-language-autonym="Fiji Hindi" data-language-local-name="Fidži hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Logaritma" title="Logaritma – fääri" lang="fo" hreflang="fo" data-title="Logaritma" data-language-autonym="Føroyskt" data-language-local-name="fääri" 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" title="Logarithme – prantsuse" lang="fr" hreflang="fr" data-title="Logarithme" data-language-autonym="Français" data-language-local-name="prantsuse" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Logartam" title="Logartam – iiri" lang="ga" hreflang="ga" data-title="Logartam" data-language-autonym="Gaeilge" data-language-local-name="iiri" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – galeegi" lang="gl" hreflang="gl" data-title="Logaritmo" data-language-autonym="Galego" data-language-local-name="galeegi" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%B0%8D%E6%95%B8" title="對數 – kani" lang="gan" hreflang="gan" data-title="對數" data-language-autonym="贛語" data-language-local-name="kani" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%9C%EA%B7%B8_(%EC%88%98%ED%95%99)" title="로그 (수학) – korea" lang="ko" hreflang="ko" data-title="로그 (수학)" data-language-autonym="한국어" data-language-local-name="korea" 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%BC%D5%B8%D5%A3%D5%A1%D6%80%D5%AB%D5%A9%D5%B4" title="Լոգարիթմ – armeenia" lang="hy" hreflang="hy" data-title="Լոգարիթմ" data-language-autonym="Հայերեն" data-language-local-name="armeenia" 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%B2%E0%A4%98%E0%A5%81%E0%A4%97%E0%A4%A3%E0%A4%95" title="लघुगणक – hindi" lang="hi" hreflang="hi" data-title="लघुगणक" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Logaritam" title="Logaritam – horvaadi" lang="hr" hreflang="hr" data-title="Logaritam" data-language-autonym="Hrvatski" data-language-local-name="horvaadi" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Logaritmo" title="Logaritmo – ido" lang="io" hreflang="io" data-title="Logaritmo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Logarithmo" title="Logarithmo – interlingua" lang="ia" hreflang="ia" data-title="Logarithmo" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Logri" title="Logri – islandi" lang="is" hreflang="is" data-title="Logri" data-language-autonym="Íslenska" data-language-local-name="islandi" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logaritmo" title="Logaritmo – itaalia" lang="it" hreflang="it" data-title="Logaritmo" data-language-autonym="Italiano" data-language-local-name="itaalia" 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" title="לוגריתם – heebrea" lang="he" hreflang="he" data-title="לוגריתם" data-language-autonym="עברית" data-language-local-name="heebrea" 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%9A%E1%83%9D%E1%83%92%E1%83%90%E1%83%A0%E1%83%98%E1%83%97%E1%83%9B%E1%83%98" title="ლოგარითმი – gruusia" lang="ka" hreflang="ka" data-title="ლოგარითმი" data-language-autonym="ქართული" data-language-local-name="gruusia" 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%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – kasahhi" lang="kk" hreflang="kk" data-title="Логарифм" data-language-autonym="Қазақша" data-language-local-name="kasahhi" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Logi" title="Logi – suahiili" lang="sw" hreflang="sw" data-title="Logi" data-language-autonym="Kiswahili" data-language-local-name="suahiili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Logaritm" title="Logaritm – Guajaana kreoolkeel" lang="gcr" hreflang="gcr" data-title="Logaritm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guajaana kreoolkeel" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logarithmus" title="Logarithmus – ladina" lang="la" hreflang="la" data-title="Logarithmus" data-language-autonym="Latina" data-language-local-name="ladina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Logaritms" title="Logaritms – läti" lang="lv" hreflang="lv" data-title="Logaritms" data-language-autonym="Latviešu" data-language-local-name="läti" 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/Logaritmas" title="Logaritmas – leedu" lang="lt" hreflang="lt" data-title="Logaritmas" data-language-autonym="Lietuvių" data-language-local-name="leedu" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Logaritmo" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Logaritm" title="Logaritm – lombardi" lang="lmo" hreflang="lmo" data-title="Logaritm" data-language-autonym="Lombard" data-language-local-name="lombardi" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu badge-Q17437796 badge-featuredarticle mw-list-item" title="eeskujulik artikkel"><a href="https://hu.wikipedia.org/wiki/Logaritmus" title="Logaritmus – ungari" lang="hu" hreflang="hu" data-title="Logaritmus" data-language-autonym="Magyar" data-language-local-name="ungari" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="eeskujulik artikkel"><a href="https://mk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Логаритам – makedoonia" lang="mk" hreflang="mk" data-title="Логаритам" data-language-autonym="Македонски" data-language-local-name="makedoonia" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Anisa" title="Anisa – malagassi" lang="mg" hreflang="mg" data-title="Anisa" data-language-autonym="Malagasy" data-language-local-name="malagassi" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B2%E0%B5%8B%E0%B4%97%E0%B4%B0%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ലോഗരിതം – malajalami" lang="ml" hreflang="ml" data-title="ലോഗരിതം" data-language-autonym="മലയാളം" data-language-local-name="malajalami" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A5%85%E0%A4%B0%E0%A4%BF%E0%A4%A6%E0%A4%AE" title="लॉगॅरिदम – marathi" lang="mr" hreflang="mr" data-title="लॉगॅरिदम" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9C%E1%80%B1%E1%80%AC%E1%80%B7%E1%80%82%E1%80%9B%E1%80%85%E1%80%BA%E1%80%9E%E1%80%99%E1%80%BA" title="လော့ဂရစ်သမ် – birma" lang="my" hreflang="my" data-title="လော့ဂရစ်သမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birma" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Logaritme" title="Logaritme – hollandi" lang="nl" hreflang="nl" data-title="Logaritme" data-language-autonym="Nederlands" data-language-local-name="hollandi" 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/%E5%AF%BE%E6%95%B0" title="対数 – jaapani" lang="ja" hreflang="ja" data-title="対数" data-language-autonym="日本語" data-language-local-name="jaapani" 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/Logaritme" title="Logaritme – norra bokmål" lang="nb" hreflang="nb" data-title="Logaritme" data-language-autonym="Norsk bokmål" data-language-local-name="norra bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Logaritme" title="Logaritme – uusnorra" lang="nn" hreflang="nn" data-title="Logaritme" data-language-autonym="Norsk nynorsk" data-language-local-name="uusnorra" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Logaritme" title="Logaritme – oksitaani" lang="oc" hreflang="oc" data-title="Logaritme" data-language-autonym="Occitan" data-language-local-name="oksitaani" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Loogarizimii" title="Loogarizimii – oromo" lang="om" hreflang="om" data-title="Loogarizimii" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Logarifm" title="Logarifm – usbeki" lang="uz" hreflang="uz" data-title="Logarifm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A8%98%E0%A9%82%E0%A8%97%E0%A8%A3%E0%A8%95" title="ਲਘੂਗਣਕ – pandžabi" lang="pa" hreflang="pa" data-title="ਲਘੂਗਣਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%84%D8%A7%DA%AF%D8%B1%D8%AA%DA%BE%D9%85" title="لاگرتھم – lahnda" lang="pnb" hreflang="pnb" data-title="لاگرتھم" data-language-autonym="پنجابی" data-language-local-name="lahnda" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Lagaridim" title="Lagaridim – Jamaica kreoolkeel" lang="jam" hreflang="jam" data-title="Lagaridim" data-language-autonym="Patois" data-language-local-name="Jamaica kreoolkeel" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Logarithmus" title="Logarithmus – alamsaksa" lang="nds" hreflang="nds" data-title="Logarithmus" data-language-autonym="Plattdüütsch" data-language-local-name="alamsaksa" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logarytm" title="Logarytm – poola" lang="pl" hreflang="pl" data-title="Logarytm" data-language-autonym="Polski" data-language-local-name="poola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="eeskujulik artikkel"><a href="https://pt.wikipedia.org/wiki/Logaritmo" title="Logaritmo – portugali" lang="pt" hreflang="pt" data-title="Logaritmo" data-language-autonym="Português" data-language-local-name="portugali" 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" title="Logaritm – rumeenia" lang="ro" hreflang="ro" data-title="Logaritm" data-language-autonym="Română" data-language-local-name="rumeenia" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="eeskujulik artikkel"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – vene" lang="ru" hreflang="ru" data-title="Логарифм" data-language-autonym="Русский" data-language-local-name="vene" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – jakuudi" lang="sah" hreflang="sah" data-title="Логарифм" data-language-autonym="Саха тыла" data-language-local-name="jakuudi" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logaritmet" title="Logaritmet – albaania" lang="sq" hreflang="sq" data-title="Logaritmet" data-language-autonym="Shqip" data-language-local-name="albaania" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Logaritmu" title="Logaritmu – sitsiilia" lang="scn" hreflang="scn" data-title="Logaritmu" data-language-autonym="Sicilianu" data-language-local-name="sitsiilia" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BD%E0%B6%9D%E0%B7%94_%E0%B6%9C%E0%B6%AB%E0%B6%9A" title="ලඝු ගණක – singali" lang="si" hreflang="si" data-title="ලඝු ගණක" data-language-autonym="සිංහල" data-language-local-name="singali" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Logarithm" title="Logarithm – lihtsustatud inglise" lang="en-simple" hreflang="en-simple" data-title="Logarithm" data-language-autonym="Simple English" data-language-local-name="lihtsustatud inglise" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Logaritmus" title="Logaritmus – slovaki" lang="sk" hreflang="sk" data-title="Logaritmus" data-language-autonym="Slovenčina" data-language-local-name="slovaki" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Logaritem" title="Logaritem – sloveeni" lang="sl" hreflang="sl" data-title="Logaritem" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" 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" title="لۆگاریتم – sorani" lang="ckb" hreflang="ckb" data-title="لۆگاریتم" data-language-autonym="کوردی" data-language-local-name="sorani" 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%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Логаритам – serbia" lang="sr" hreflang="sr" data-title="Логаритам" data-language-autonym="Српски / srpski" data-language-local-name="serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh badge-Q70893996 mw-list-item" title=""><a href="https://sh.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Логаритам – serbia-horvaadi" lang="sh" hreflang="sh" data-title="Логаритам" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbia-horvaadi" 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/Logaritmi" title="Logaritmi – soome" lang="fi" hreflang="fi" data-title="Logaritmi" data-language-autonym="Suomi" data-language-local-name="soome" 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/Logaritm" title="Logaritm – rootsi" lang="sv" hreflang="sv" data-title="Logaritm" data-language-autonym="Svenska" data-language-local-name="rootsi" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – tagalogi" lang="tl" hreflang="tl" data-title="Logaritmo" data-language-autonym="Tagalog" data-language-local-name="tagalogi" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AE%9F%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%88" title="மடக்கை – tamili" lang="ta" hreflang="ta" data-title="மடக்கை" data-language-autonym="தமிழ்" data-language-local-name="tamili" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – tatari" lang="tt" hreflang="tt" data-title="Логарифм" data-language-autonym="Татарча / tatarça" data-language-local-name="tatari" class="interlanguage-link-target"><span>Татарча / tatarça</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" title="ลอการิทึม – tai" lang="th" hreflang="th" data-title="ลอการิทึม" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="eeskujulik artikkel"><a href="https://vi.wikipedia.org/wiki/Logarit" title="Logarit – vietnami" lang="vi" hreflang="vi" data-title="Logarit" data-language-autonym="Tiếng Việt" data-language-local-name="vietnami" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Logaritma" title="Logaritma – türgi" lang="tr" hreflang="tr" data-title="Logaritma" data-language-autonym="Türkçe" data-language-local-name="türgi" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – ukraina" lang="uk" hreflang="uk" data-title="Логарифм" data-language-autonym="Українська" data-language-local-name="ukraina" 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%84%D8%A7%DA%AF%D8%B1%D8%AA%DA%BE%D9%85" title="لاگرتھم – urdu" lang="ur" hreflang="ur" data-title="لاگرتھم" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Logaritmo" title="Logaritmo – varai" lang="war" hreflang="war" data-title="Logaritmo" data-language-autonym="Winaray" data-language-local-name="varai" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AF%B9%E6%95%B0" title="对数 – uu" lang="wuu" hreflang="wuu" data-title="对数" data-language-autonym="吴语" data-language-local-name="uu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%9C%D7%90%D7%92%D7%90%D7%A8%D7%99%D7%98%D7%9D" title="לאגאריטם – jidiši" lang="yi" hreflang="yi" data-title="לאגאריטם" data-language-autonym="ייִדיש" data-language-local-name="jidiši" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B0%8D%E6%95%B8" title="對數 – kantoni" lang="yue" hreflang="yue" data-title="對數" data-language-autonym="粵語" data-language-local-name="kantoni" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Logaritma" title="Logaritma – dõmli" lang="diq" hreflang="diq" data-title="Logaritma" data-language-autonym="Zazaki" data-language-local-name="dõmli" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Luogar%C4%97tmos" title="Luogarėtmos – žemaidi" lang="sgs" hreflang="sgs" data-title="Luogarėtmos" data-language-autonym="Žemaitėška" data-language-local-name="žemaidi" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AF%B9%E6%95%B0" title="对数 – hiina" lang="zh" hreflang="zh" data-title="对数" data-language-autonym="中文" data-language-local-name="hiina" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11197#sitelinks-wikipedia" title="Muuda keelelinke" class="wbc-editpage">Muuda linke</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Nimeruumid"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Logaritm" title="Vaata sisulehekülge [c]" accesskey="c"><span>Artikkel</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Arutelu:Logaritm&action=edit&redlink=1" rel="discussion" class="new" title="Arutelu selle lehekülje sisu kohta (pole veel kirjutatud) [t]" accesskey="t"><span>Arutelu</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" 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mw-list-item"><a href="/w/index.php?title=Logaritm&action=edit" title="Redigeeri selle lehekülje lähteteksti [e]" accesskey="e"><span>Muuda lähteteksti</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Logaritm&action=history"><span>Näita ajalugu</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Üldine </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Eri:Lingid_siia/Logaritm" title="Kõik viki leheküljed, mis siia viitavad [j]" accesskey="j"><span>Lingid siia</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Eri:Seotud_muudatused/Logaritm" rel="nofollow" title="Viimased muudatused lehekülgedel, millele on siit viidatud [k]" accesskey="k"><span>Seotud muudatused</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Eri:Erilehek%C3%BCljed" title="Erilehekülgede loend [q]" accesskey="q"><span>Erileheküljed</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Logaritm&oldid=6323722" title="Püsilink lehekülje sellele redaktsioonile"><span>Püsilink</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Logaritm&action=info" title="Lisateave selle lehekülje kohta"><span>Lehekülje teave</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Eri:Sellele_lehek%C3%BCljele_viitamine&page=Logaritm&id=6323722&wpFormIdentifier=titleform" title="Teave sellele leheküljele viitamise kohta"><span>Viita leheküljele</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Eri:UrlShortener&url=https%3A%2F%2Fet.wikipedia.org%2Fwiki%2FLogaritm"><span>Hangi lühendatud URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a 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.mw-parser-output .cmbox{background-color:#0d1a27}html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cmbox{background-color:#0d1a27}html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}}</style></div><table class="metadata plainlinks ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></span></span></td><td class="mbox-text"><div class="mbox-text-span"><b>Artiklis ei ole piisavalt <a href="/wiki/Vikipeedia:Viitamine" title="Vikipeedia:Viitamine">viiteid</a></b>.<span class="hide-when-compact"></span><span class="hide-when-compact"> <br /><small>Palun aita väiteid kontrollida ja <a class="external text" href="https://et.wikipedia.org/w/index.php?title=Logaritm&action=edit">viiteid lisada</a></small>. <small>(<a href="/wiki/Juhend:Toimetusm%C3%A4rkuste_eemaldamine" title="Juhend:Toimetusmärkuste eemaldamine">Kuidas ja millal see märkus eemaldada?</a>)</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fail:Binary_logarithm_plot_with_ticks.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/220px-Binary_logarithm_plot_with_ticks.svg.png" decoding="async" width="220" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/330px-Binary_logarithm_plot_with_ticks.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/440px-Binary_logarithm_plot_with_ticks.svg.png 2x" data-file-width="408" data-file-height="325" /></a><figcaption><a href="/wiki/Graafik" class="mw-disambig" title="Graafik">Graafik</a>, mis kujutab logaritmi alusel 2: y=log<sub>2</sub>x</figcaption></figure> <p><a href="/wiki/Arv" title="Arv">Arv</a> <i>x</i> on arvu <i>b</i> <b>logaritm alusel</b> <i>a</i>, kui <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 b=a^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2377177f73c768b94712406a217859bcc41173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.498ex; height:2.343ex;" alt="{\displaystyle b=a^{x}}"></span> ehk arv <i>x</i> on <a href="/wiki/Astendaja" class="mw-redirect" title="Astendaja">astendaja</a> (eksponent), millega arvu <i>a</i> <a href="/wiki/Astendamine" title="Astendamine">astendades</a> on tulemuseks arv <i>b</i> ehk </p> <dl><dd><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 _{a}{b}=x\,\iff \,b=a^{x}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{a}{b}=x\,\iff \,b=a^{x}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf2a7db18f323f89d88e00462682b975daa515b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.284ex; height:2.843ex;" alt="{\displaystyle \log _{a}{b}=x\,\iff \,b=a^{x}\ .}"></span></dd></dl> <p>Kui <i>a</i> ja <i>b</i> on positiivsed <a href="/wiki/Reaalarv" title="Reaalarv">reaalarvud</a>, siis on log<sub><i>a</i></sub><i>b</i> reaalarv. Aluse <i>a</i> väärtus peab olema kas <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<a<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<a<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfd02fa239e3f0b03bda321fdc6f309e6b1d915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.752ex; height:2.176ex;" alt="{\displaystyle 0<a<1}"></span> või <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 a>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5b9d9fb0ff9d4455e75ccd29676bd7f33da80e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>1}"></span>; tavaliselt kasutatakse alustena arve 10, <a href="/wiki/E_(arv)" title="E (arv)"><i>e</i></a> või 2. Logaritmi alusel <a href="/wiki/E_(arv)" title="E (arv)"><i>e</i></a> nimetatakse <a href="/wiki/Naturaallogaritm" title="Naturaallogaritm">naturaallogaritmiks</a>. Logaritmid on defineeritud reaalarvudele ja <a href="/wiki/Kompleksarv" title="Kompleksarv">kompleksarvudele</a>. Naturaallogaritmi pöördfunktsioon on <a href="/wiki/Eksponentfunktsioon" title="Eksponentfunktsioon">eksponentfunktsioon</a> ehk eksponentsiaalfunktsioon. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Logaritmi_omadused">Logaritmi omadused</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritm&veaction=edit&section=1" title="Muuda alaosa "Logaritmi omadused"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritm&action=edit&section=1" title="Muuda alaosa "Logaritmi omadused" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><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 _{a}{1}=0\ ,\quad {\text{ sest }}\quad a^{0}=1\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> sest </mtext> </mrow> <mspace width="1em" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{a}{1}=0\ ,\quad {\text{ sest }}\quad a^{0}=1\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4623427e196e8da3d94d644d29ca37547b7e68cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.847ex; height:3.176ex;" alt="{\displaystyle \log _{a}{1}=0\ ,\quad {\text{ sest }}\quad a^{0}=1\ .}"></span></dd> <dd><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 _{a}{a}=1\ ,\quad {\text{ sest }}\quad a^{1}=a\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> sest </mtext> </mrow> <mspace width="1em" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>a</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{a}{a}=1\ ,\quad {\text{ sest }}\quad a^{1}=a\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76914eaf6087f7d2432e5971c34d8a53bf825df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.982ex; height:3.176ex;" alt="{\displaystyle \log _{a}{a}=1\ ,\quad {\text{ sest }}\quad a^{1}=a\ .}"></span></dd></dl> <p>log<sub>a</sub>0 ei ole määratud kuna pole olemas sellist arvu<i> x</i> , mille puhul <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 a^{x}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{x}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b09e322edb67407ef37fd007cb64032dbee5035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.663ex; height:2.343ex;" alt="{\displaystyle a^{x}=0}"></span>. <a href="/w/index.php?title=Logaritmfunktsioon&action=edit&redlink=1" class="new" title="Logaritmfunktsioon (pole veel kirjutatud)">Logaritmfunktsiooni</a> <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)=log_{a}{x}}"> <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> <mi>l</mi> <mi>o</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=log_{a}{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdd930aceadb01c939aaf0836f557e0d9966864" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.878ex; height:2.843ex;" alt="{\displaystyle f(x)=log_{a}{x}}"></span> graafikul on vertikaalne <a href="/wiki/As%C3%BCmptoot" title="Asümptoot">asümptoot</a> <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>. </p> <div class="mw-heading mw-heading3"><h3 id="Logaritm_korrutisest_ja_jagatisest">Logaritm korrutisest ja jagatisest</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritm&veaction=edit&section=2" title="Muuda alaosa "Logaritm korrutisest ja jagatisest"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritm&action=edit&section=2" title="Muuda alaosa "Logaritm korrutisest ja jagatisest" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Logaritmi tähtsamaid omadusi on </p> <dl><dd><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 _{a}{bc}=\log _{a}{b}+\log _{a}{c}\ ,\quad {\text{ sest }}\quad a^{m}\times a^{n}=a^{m+n}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mi>c</mi> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> sest </mtext> </mrow> <mspace width="1em" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>×</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{a}{bc}=\log _{a}{b}+\log _{a}{c}\ ,\quad {\text{ sest }}\quad a^{m}\times a^{n}=a^{m+n}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08ecf242fb6d2ed188743d67a4fda3f358dd61b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.21ex; height:3.009ex;" alt="{\displaystyle \log _{a}{bc}=\log _{a}{b}+\log _{a}{c}\ ,\quad {\text{ sest }}\quad a^{m}\times a^{n}=a^{m+n}\ ,}"></span> kui <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 b=a^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771ba37b99ff2693e41d9a63b3319727a1200c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.001ex; height:2.343ex;" alt="{\displaystyle b=a^{m}}"></span> ja <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 c=a^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=a^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8133171cf6d7059b3e53e5954715316e1fb0a7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.554ex; height:2.343ex;" alt="{\displaystyle c=a^{n}}"></span> .</dd></dl> <p>Samuti </p> <dl><dd><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 _{a}{\frac {b}{c}}=\log _{a}{b}-\log _{a}{c}\ ,\quad {\text{ sest }}\quad {\frac {a^{m}}{a^{n}}}=a^{m-n}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>c</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> sest </mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{a}{\frac {b}{c}}=\log _{a}{b}-\log _{a}{c}\ ,\quad {\text{ sest }}\quad {\frac {a^{m}}{a^{n}}}=a^{m-n}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44c3bdf1a74f9d96f0ce70bc995cf2ed61c2080d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.596ex; height:5.343ex;" alt="{\displaystyle \log _{a}{\frac {b}{c}}=\log _{a}{b}-\log _{a}{c}\ ,\quad {\text{ sest }}\quad {\frac {a^{m}}{a^{n}}}=a^{m-n}\ ,}"></span> kui <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 b=a^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/771ba37b99ff2693e41d9a63b3319727a1200c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.001ex; height:2.343ex;" alt="{\displaystyle b=a^{m}}"></span> ja <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 c=a^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=a^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8133171cf6d7059b3e53e5954715316e1fb0a7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.554ex; height:2.343ex;" alt="{\displaystyle c=a^{n}}"></span> .</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Logaritm_astmest">Logaritm astmest</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritm&veaction=edit&section=3" title="Muuda alaosa "Logaritm astmest"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritm&action=edit&section=3" title="Muuda alaosa "Logaritm astmest" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tähtis omadus on ka astendamise taandumine <a href="/wiki/Korrutamine" title="Korrutamine">korrutamiseks</a>. Logaritmi definitsiooni kohaselt on logaritm arvust <i>b</i> alusel <i>a</i> arv, millega arvu <i>a</i> astendades on tulemuseks arv <i>b</i>: </p> <dl><dd><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 b=a^{\log _{a}{b}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a^{\log _{a}{b}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2055c50e367fc446ec319cfc715d628d23b89d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.85ex; height:2.676ex;" alt="{\displaystyle b=a^{\log _{a}{b}}\ .}"></span></dd></dl> <p>Astendades <a href="/wiki/V%C3%B5rrand" title="Võrrand">võrrandi</a> mõlemad pooled arvuga <i>n</i> : </p> <dl><dd><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 b^{n}=\left(a^{\log _{a}{b}}\right)^{n}=a^{n\log _{a}{b}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </mrow> </msup> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{n}=\left(a^{\log _{a}{b}}\right)^{n}=a^{n\log _{a}{b}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420318871ca6d2a645dfb46bcf39540778082c98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.414ex; height:3.509ex;" alt="{\displaystyle b^{n}=\left(a^{\log _{a}{b}}\right)^{n}=a^{n\log _{a}{b}}\ ,}"></span></dd></dl> <p>ja võttes logaritmid, on tulemuseks </p> <dl><dd><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 _{a}\left(b^{n}\right)=n\log _{a}{b}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mi>n</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{a}\left(b^{n}\right)=n\log _{a}{b}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6b4234b78cc548186810548f6bfdffc3cc95d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.665ex; height:2.843ex;" alt="{\displaystyle \log _{a}\left(b^{n}\right)=n\log _{a}{b}\ .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Logaritmi_aluse_vahetamine">Logaritmi aluse vahetamine</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritm&veaction=edit&section=4" title="Muuda alaosa "Logaritmi aluse vahetamine"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritm&action=edit&section=4" title="Muuda alaosa "Logaritmi aluse vahetamine" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Kalkulaator" title="Kalkulaator">Kalkulaatoriga</a> töötamisel on kasulik teada, et peale olemasolevate alustega logaritmfunktsioonide (tavaliselt ln <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> ja log <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>) saab leida logaritmi ka muudel alustel, kasutades omadust </p> <dl><dd><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 _{a}{b}={\frac {\log _{c}{b}}{\log _{c}{a}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{a}{b}={\frac {\log _{c}{b}}{\log _{c}{a}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22f959a64884d85a4fa198ef848a1bdb19836bc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.573ex; height:6.176ex;" alt="{\displaystyle \log _{a}{b}={\frac {\log _{c}{b}}{\log _{c}{a}}}.}"></span></dd></dl> <p>Näiteks <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 _{2}{5}={\frac {\log _{10}5}{\log _{10}2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>5</mn> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}{5}={\frac {\log _{10}5}{\log _{10}2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/999946bc29675aa22fbe61b6d67dc3da159caad6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.555ex; height:6.176ex;" alt="{\displaystyle \log _{2}{5}={\frac {\log _{10}5}{\log _{10}2}}.}"></span> </p><p>See tuleneb sellest, et logaritmi definitsiooni kohaselt </p> <dl><dd><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 b=a^{x}\,\iff \,x=\log _{a}{b}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a^{x}\,\iff \,x=\log _{a}{b}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b238ff30684f24ba866f80cd7d31302c649e4c40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.09ex; height:2.843ex;" alt="{\displaystyle b=a^{x}\,\iff \,x=\log _{a}{b}\,,}"></span></dd></dl> <p>kuid <i>x</i> saab leida ka kasutades alust <i>c</i> : </p> <dl><dd><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 _{c}{b}=\log _{c}{a^{x}}=x\log _{c}{a}\iff {\frac {\log _{c}{b}}{\log _{c}{a}}}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mrow> <mo>=</mo> <mi>x</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{c}{b}=\log _{c}{a^{x}}=x\log _{c}{a}\iff {\frac {\log _{c}{b}}{\log _{c}{a}}}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615930809cbbbb1d18cc426d3162729c3b26a23e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.147ex; height:6.176ex;" alt="{\displaystyle \log _{c}{b}=\log _{c}{a^{x}}=x\log _{c}{a}\iff {\frac {\log _{c}{b}}{\log _{c}{a}}}=x}"></span></dd></dl> <p>kus a ≠ 1, sest log<sub><i>c</i></sub> 1 = 0. Iga arv astmes 0 on 1. </p> <div class="mw-heading mw-heading2"><h2 id="Logaritmi_rakendused">Logaritmi rakendused</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritm&veaction=edit&section=5" title="Muuda alaosa "Logaritmi rakendused"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritm&action=edit&section=5" title="Muuda alaosa "Logaritmi rakendused" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mõned logaritmi praktilised rakendused. </p> <ul><li><a href="/wiki/Keemia" title="Keemia">Keemias</a> on <a href="/wiki/H%C3%BCdrooniumioon" title="Hüdrooniumioon">hüdrooniumiooni</a> H<sub>3</sub>O<sup>+</sup> (H<sup>+</sup> <a href="/wiki/Ioon" title="Ioon">iooni</a> esinemine <a href="/wiki/Vesi" title="Vesi">vees</a>) keemilise aktiivsuse logaritm alusel 10 vastandväärtus tuntud kui <a href="/wiki/PH" class="mw-redirect" title="PH">pH</a>. H<sub>3</sub>O<sup>+</sup> ioonide aktiivsus neutraalses vees on 10<sup>−7</sup> <a href="/wiki/Mool" title="Mool">mooli</a> <a href="/wiki/Liiter" title="Liiter">liitris</a> <a href="/wiki/Temperatuur" title="Temperatuur">temperatuuril</a> 25 °C, ehk pH 7.</li> <li><a href="/wiki/Bell" title="Bell">Bell</a> on <a href="/wiki/M%C3%B5%C3%B5t%C3%BChik" title="Mõõtühik">mõõtühik</a>, mis on suhte logaritm alusel 10, näiteks <a href="/wiki/V%C3%B5imsus" title="Võimsus">võimsuste</a> ja <a href="/wiki/Elektripinge" class="mw-redirect" title="Elektripinge">elektripingete</a> tasemete suhted. Seda kasutatakse <a href="/wiki/Telekommunikatsioon" title="Telekommunikatsioon">telekommunikatsioonis</a>, <a href="/wiki/Elektroonika" title="Elektroonika">elektroonikas</a> ja <a href="/wiki/Akustika" title="Akustika">akustikas</a>. Bell on nimetatud <a href="/wiki/Alexander_Graham_Bell" title="Alexander Graham Bell">Alexander Graham Belli</a> järgi. <a href="/wiki/Detsibell" title="Detsibell">Detsibelli</a> (0,1 belli) kasutatakse sagedamini.</li> <li><a href="/wiki/Maav%C3%A4rin" title="Maavärin">Maavärina</a> võimsuse <a href="/wiki/Richteri_skaala" title="Richteri skaala">Richteri skaala</a> mõõdab maavärina intensiivsust logaritmilisel skaalal.</li> <li><a href="/w/index.php?title=Spektromeetria&action=edit&redlink=1" class="new" title="Spektromeetria (pole veel kirjutatud)">Spektromeetrias</a> ja <a href="/wiki/Optika" title="Optika">optikas</a> mõõdetakse <a href="/w/index.php?title=Optiline_tihedus&action=edit&redlink=1" class="new" title="Optiline tihedus (pole veel kirjutatud)">optilist tihedust</a> ühikuga, mis on seotud logaritmiga.</li> <li><a href="/wiki/Astronoomia" title="Astronoomia">Astronoomias</a> on <a href="/wiki/N%C3%A4iv_t%C3%A4hesuurus" class="mw-redirect" title="Näiv tähesuurus">näiv tähesuurus</a> <a href="/wiki/Taevakeha" title="Taevakeha">taevakeha</a> <a href="/wiki/N%C3%A4iv_heledus" class="mw-redirect" title="Näiv heledus">näivat heledust</a> logaritmiliselt väljendav arv, kuna <a href="/wiki/Silm" title="Silm">silm</a> reageerib heledusele ligikaudu logaritmiliselt.</li> <li><a href="/wiki/Ps%C3%BChhof%C3%BC%C3%BCsika" title="Psühhofüüsika">Psühhofüüsikas</a> väljendab <a href="/w/index.php?title=Weberi-Fechneri_seadus&action=edit&redlink=1" class="new" title="Weberi-Fechneri seadus (pole veel kirjutatud)">Weberi-Fechneri seadus</a> logaritmilist seost <a href="/wiki/Stiimul" title="Stiimul">stiimuli</a> ja <a href="/wiki/Taju" title="Taju">taju</a> vahel (<a href="/w/index.php?title=Stevensi_seadus&action=edit&redlink=1" class="new" title="Stevensi seadus (pole veel kirjutatud)">Stevensi seadus</a> on siiski täpsem).</li> <li><a href="/wiki/Informaatika" title="Informaatika">Informaatikas</a> esinevad logaritmid seostena <a href="/wiki/Algoritmiline_keerukus" title="Algoritmiline keerukus">algoritmilises keerukuses</a>. Näiteks <i>N</i> elemendi <a href="/wiki/Sorteerimine" title="Sorteerimine">järjestamine</a> kasutades võrdlust, võib aega võtta proportsionaalselt <i>N</i> × log<sub>10</sub> <i>N</i>. Sarnaselt kasutatakse logaritme alusel 2, et väljendada andmemahtu või <a href="/wiki/M%C3%A4lu_(arvuti)" title="Mälu (arvuti)">mälu</a>, mis on vaja selleks, et esitada arvu <a href="/wiki/Kahends%C3%BCsteem" title="Kahendsüsteem">kahendsüsteemis</a> – <i>k</i> biti abil saab väljendada 2<sup><i>k</i></sup> erinevat väärtust, seega iga <a href="/wiki/Naturaalarv" title="Naturaalarv">naturaalarvu</a> <i>N</i> saab väljendada kuni (log<sub>2</sub> <i>N</i>) + 1 bitiga.</li> <li><a href="/wiki/Informatsiooniteooria" title="Informatsiooniteooria">Informatsiooniteoorias</a> kasutatakse logaritme info hulga mõõtmiseks. Kui sõnumi vastuvõtjal on oodata mistahes sõnumit <i>N</i> võimalikust sama <a href="/wiki/T%C3%B5en%C3%A4osus" title="Tõenäosus">tõenäosusega</a> sõnumist, siis iga sõnumiga saadud info hulk on log<sub>2</sub> <i>N</i> <a href="/wiki/Bitt" title="Bitt">bitti</a>.</li> <li><a href="/wiki/Intervall_(muusika)" title="Intervall (muusika)">Muusikalisi intervalle</a> mõõdetakse logaritmiliselt <a href="/wiki/Pooltoon" title="Pooltoon">pooltoonidena</a>. Kahe noodi vaheline intervall pooltoonides on logaritm alusel 2<sup>1/12</sup> sageduse suhtest. (Vaata ka <a href="/wiki/MIDI" title="MIDI">MIDI</a>.)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Vaata_ka">Vaata ka</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritm&veaction=edit&section=6" title="Muuda alaosa "Vaata ka"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritm&action=edit&section=6" title="Muuda alaosa "Vaata ka" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Naturaallogaritm" title="Naturaallogaritm">Naturaallogaritm</a></li> <li><a href="/wiki/Logaritmiline_diferentseerimine" title="Logaritmiline diferentseerimine">Logaritmiline diferentseerimine</a></li> <li><a href="/wiki/Logaritmiline_skaala" title="Logaritmiline skaala">Logaritmiline skaala</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Välislingid"><span id="V.C3.A4lislingid"></span>Välislingid</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritm&veaction=edit&section=7" title="Muuda alaosa "Välislingid"" class="mw-editsection-visualeditor"><span>muuda</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritm&action=edit&section=7" title="Muuda alaosa "Välislingid" lähteteksti"><span>muuda lähteteksti</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>[<a rel="nofollow" class="external autonumber" href="https://www.youtube.com/watch?v=vzV50goW_WM">[1]</a>], [<a rel="nofollow" class="external autonumber" href="https://www.youtube.com/watch?v=VRzH4xB0GdM">[2]</a>] Professor Roger Bowley selgitab logaritme <a href="/wiki/YouTube" title="YouTube">YouTube</a>`i kanalis Numberphile.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Pärit leheküljelt "<a dir="ltr" href="https://et.wikipedia.org/w/index.php?title=Logaritm&oldid=6323722">https://et.wikipedia.org/w/index.php?title=Logaritm&oldid=6323722</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Eri:Kategooriad" title="Eri:Kategooriad">Kategooria</a>: <ul><li><a href="/wiki/Kategooria:Funktsioonid" title="Kategooria:Funktsioonid">Funktsioonid</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Peidetud kategooria: <ul><li><a href="/wiki/Kategooria:Ebapiisavalt_viidatud_artiklid" title="Kategooria:Ebapiisavalt viidatud artiklid">Ebapiisavalt viidatud artiklid</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Selle lehekülje viimane muutmine: 09:08, 15. veebruar 2023.</li> <li id="footer-info-copyright">Tekst on kasutatav vastavalt Creative Commonsi litsentsile "<a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.et">Autorile viitamine + jagamine samadel tingimustel</a>"; sellele võivad lisanduda täiendavad tingimused. 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