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class="vector-toc-link" href="#Historique"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Historique</span> </div> </a> <ul id="toc-Historique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriétés_des_fonctions_logarithmes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propriétés_des_fonctions_logarithmes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Propriétés des fonctions logarithmes</span> </div> </a> <button aria-controls="toc-Propriétés_des_fonctions_logarithmes-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>Afficher / masquer la sous-section Propriétés des fonctions logarithmes</span> </button> <ul id="toc-Propriétés_des_fonctions_logarithmes-sublist" class="vector-toc-list"> <li id="toc-Propriétés_algébriques" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriétés_algébriques"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Propriétés algébriques</span> </div> </a> <ul id="toc-Propriétés_algébriques-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Changement_de_base" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Changement_de_base"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Changement de base</span> </div> </a> <ul id="toc-Changement_de_base-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dérivée" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dérivée"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Dérivée</span> </div> </a> <ul id="toc-Dérivée-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nombre_de_chiffres_avant_la_virgule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nombre_de_chiffres_avant_la_virgule"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Nombre de chiffres avant la virgule</span> </div> </a> <ul id="toc-Nombre_de_chiffres_avant_la_virgule-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fonction_réciproque_(antilogarithme)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fonction_réciproque_(antilogarithme)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Fonction réciproque (antilogarithme)</span> </div> </a> <ul id="toc-Fonction_réciproque_(antilogarithme)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fonctions_logarithme_courantes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fonctions_logarithme_courantes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Fonctions logarithme courantes</span> </div> </a> <button aria-controls="toc-Fonctions_logarithme_courantes-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>Afficher / masquer la sous-section Fonctions logarithme courantes</span> </button> <ul id="toc-Fonctions_logarithme_courantes-sublist" class="vector-toc-list"> <li id="toc-Logarithme_népérien" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithme_népérien"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Logarithme népérien</span> </div> </a> <ul id="toc-Logarithme_népérien-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logarithme_décimal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithme_décimal"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Logarithme décimal</span> </div> </a> <ul id="toc-Logarithme_décimal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logarithme_binaire" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logarithme_binaire"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Logarithme binaire</span> </div> </a> <ul id="toc-Logarithme_binaire-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cologarithme" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cologarithme"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Cologarithme</span> </div> </a> <ul id="toc-Cologarithme-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Généralisations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Généralisations"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Généralisations</span> </div> </a> <ul id="toc-Généralisations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes et références</span> </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Voir aussi</span> </div> </a> <button aria-controls="toc-Voir_aussi-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>Afficher / masquer la sous-section Voir aussi</span> </button> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> <li id="toc-Applications_pratiques" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Applications_pratiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1.1</span> <span>Applications pratiques</span> </div> </a> <ul id="toc-Applications_pratiques-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Liens_externes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liens_externes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Liens externes</span> </div> </a> <ul id="toc-Liens_externes-sublist" class="vector-toc-list"> </ul> </li> </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="Sommaire" 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="Basculer la table des matières" > <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">Basculer la table des matières</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">Logarithme</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="Aller à un article dans une autre langue. Disponible en 109 langues." > <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 langues</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 – afrikaans" lang="af" hreflang="af" data-title="Logaritme" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" 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 – alémanique" lang="gsw" hreflang="gsw" data-title="Logarithmus" data-language-autonym="Alemannisch" data-language-local-name="alémanique" 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 – amharique" lang="am" hreflang="am" data-title="ሎጋሪዝም" data-language-autonym="አማርኛ" data-language-local-name="amharique" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Logaritmo" title="Logaritmo – aragonais" lang="an" hreflang="an" data-title="Logaritmo" data-language-autonym="Aragonés" data-language-local-name="aragonais" class="interlanguage-link-target"><span>Aragonés</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 – arabe" lang="ar" hreflang="ar" data-title="لوغاريتم" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</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 – arabe marocain" lang="ary" hreflang="ary" data-title="لوڭاريتم" data-language-autonym="الدارجة" data-language-local-name="arabe marocain" class="interlanguage-link-target"><span>الدارجة</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 – assamais" lang="as" hreflang="as" data-title="ঘাতাংক" data-language-autonym="অসমীয়া" data-language-local-name="assamais" 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 – asturien" lang="ast" hreflang="ast" data-title="Logaritmu" data-language-autonym="Asturianu" data-language-local-name="asturien" 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 – azerbaïdjanais" lang="az" hreflang="az" data-title="Loqarifm" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" class="interlanguage-link-target"><span>Azərbaycanca</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 – bachkir" lang="ba" hreflang="ba" data-title="Логарифм" data-language-autonym="Башҡортса" data-language-local-name="bachkir" class="interlanguage-link-target"><span>Башҡортса</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 – samogitien" lang="sgs" hreflang="sgs" data-title="Luogarėtmos" data-language-autonym="Žemaitėška" data-language-local-name="samogitien" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Logaritmo" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</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 – biélorusse" lang="be" hreflang="be" data-title="Лагарыфм" data-language-autonym="Беларуская" data-language-local-name="biélorusse" 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 – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лягарытм" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</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 – bulgare" lang="bg" hreflang="bg" data-title="Логаритъм" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Logaritma" title="Logaritma – banjar" lang="bjn" hreflang="bjn" data-title="Logaritma" data-language-autonym="Banjar" data-language-local-name="banjar" class="interlanguage-link-target"><span>Banjar</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-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Logaritm" title="Logaritm – breton" lang="br" hreflang="br" data-title="Logaritm" data-language-autonym="Brezhoneg" data-language-local-name="breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Logaritam" title="Logaritam – bosniaque" lang="bs" hreflang="bs" data-title="Logaritam" data-language-autonym="Bosanski" data-language-local-name="bosniaque" class="interlanguage-link-target"><span>Bosanski</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 – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Логарифм" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" 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 – catalan" lang="ca" hreflang="ca" data-title="Logaritme" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</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-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Logaritmus" title="Logaritmus – tchèque" lang="cs" hreflang="cs" data-title="Logaritmus" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</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 – tchouvache" lang="cv" hreflang="cv" data-title="Логарифм" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Logarithm" title="Logarithm – gallois" lang="cy" hreflang="cy" data-title="Logarithm" data-language-autonym="Cymraeg" data-language-local-name="gallois" 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 – danois" lang="da" hreflang="da" data-title="Logaritme" data-language-autonym="Dansk" data-language-local-name="danois" 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" title="Logarithmus – allemand" lang="de" hreflang="de" data-title="Logarithmus" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Logaritma" title="Logaritma – Zazaki" lang="diq" hreflang="diq" data-title="Logaritma" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</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 – grec" lang="el" hreflang="el" data-title="Λογάριθμος" data-language-autonym="Ελληνικά" data-language-local-name="grec" 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 – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Logarìtem" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualité"><a href="https://en.wikipedia.org/wiki/Logarithm" title="Logarithm – anglais" lang="en" hreflang="en" data-title="Logarithm" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Logaritmo" title="Logaritmo – espéranto" lang="eo" hreflang="eo" data-title="Logaritmo" data-language-autonym="Esperanto" data-language-local-name="espéranto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Logaritmo" title="Logaritmo – espagnol" lang="es" hreflang="es" data-title="Logaritmo" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Logaritm" title="Logaritm – estonien" lang="et" hreflang="et" data-title="Logaritm" data-language-autonym="Eesti" data-language-local-name="estonien" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Logaritmo" title="Logaritmo – basque" lang="eu" hreflang="eu" data-title="Logaritmo" data-language-autonym="Euskara" data-language-local-name="basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Logaritmu" title="Logaritmu – estrémègne" lang="ext" hreflang="ext" data-title="Logaritmu" data-language-autonym="Estremeñu" data-language-local-name="estrémègne" class="interlanguage-link-target"><span>Estremeñu</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 – persan" lang="fa" hreflang="fa" data-title="لگاریتم" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Logaritmi" title="Logaritmi – finnois" lang="fi" hreflang="fi" data-title="Logaritmi" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Logaritma" title="Logaritma – féroïen" lang="fo" hreflang="fo" data-title="Logaritma" data-language-autonym="Føroyskt" data-language-local-name="féroïen" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Logartam" title="Logartam – irlandais" lang="ga" hreflang="ga" data-title="Logartam" data-language-autonym="Gaeilge" data-language-local-name="irlandais" class="interlanguage-link-target"><span>Gaeilge</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 – gan" lang="gan" hreflang="gan" data-title="對數" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Logaritm" title="Logaritm – créole guyanais" lang="gcr" hreflang="gcr" data-title="Logaritm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="créole guyanais" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – galicien" lang="gl" hreflang="gl" data-title="Logaritmo" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target"><span>Galego</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 – hébreu" lang="he" hreflang="he" data-title="לוגריתם" data-language-autonym="עברית" data-language-local-name="hébreu" class="interlanguage-link-target"><span>עברית</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-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Logarithm" title="Logarithm – hindi fidjien" lang="hif" hreflang="hif" data-title="Logarithm" data-language-autonym="Fiji Hindi" data-language-local-name="hindi fidjien" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Logaritam" title="Logaritam – croate" lang="hr" hreflang="hr" data-title="Logaritam" data-language-autonym="Hrvatski" data-language-local-name="croate" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualité"><a href="https://hu.wikipedia.org/wiki/Logaritmus" title="Logaritmus – hongrois" lang="hu" hreflang="hu" data-title="Logaritmus" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</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 – arménien" lang="hy" hreflang="hy" data-title="Լոգարիթմ" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target"><span>Հայերեն</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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logaritma" title="Logaritma – indonésien" lang="id" hreflang="id" data-title="Logaritma" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Logri" title="Logri – islandais" lang="is" hreflang="is" data-title="Logri" data-language-autonym="Íslenska" data-language-local-name="islandais" 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 – italien" lang="it" hreflang="it" data-title="Logaritmo" data-language-autonym="Italiano" data-language-local-name="italien" class="interlanguage-link-target"><span>Italiano</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 – japonais" lang="ja" hreflang="ja" data-title="対数" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Lagaridim" title="Lagaridim – créole jamaïcain" lang="jam" hreflang="jam" data-title="Lagaridim" data-language-autonym="Patois" data-language-local-name="créole jamaïcain" class="interlanguage-link-target"><span>Patois</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 – géorgien" lang="ka" hreflang="ka" data-title="ლოგარითმი" data-language-autonym="ქართული" data-language-local-name="géorgien" 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 – kazakh" lang="kk" hreflang="kk" data-title="Логарифм" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</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) – coréen" lang="ko" hreflang="ko" data-title="로그 (수학)" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logarithmus" title="Logarithmus – latin" lang="la" hreflang="la" data-title="Logarithmus" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</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 – lombard" lang="lmo" hreflang="lmo" data-title="Logaritm" data-language-autonym="Lombard" data-language-local-name="lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Logaritmas" title="Logaritmas – lituanien" lang="lt" hreflang="lt" data-title="Logaritmas" data-language-autonym="Lietuvių" data-language-local-name="lituanien" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Logaritms" title="Logaritms – letton" lang="lv" hreflang="lv" data-title="Logaritms" data-language-autonym="Latviešu" data-language-local-name="letton" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Anisa" title="Anisa – malgache" lang="mg" hreflang="mg" data-title="Anisa" data-language-autonym="Malagasy" data-language-local-name="malgache" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualité"><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 – macédonien" lang="mk" hreflang="mk" data-title="Логаритам" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</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 – malayalam" lang="ml" hreflang="ml" data-title="ലോഗരിതം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" 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-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logaritma" title="Logaritma – malais" lang="ms" hreflang="ms" data-title="Logaritma" data-language-autonym="Bahasa Melayu" data-language-local-name="malais" class="interlanguage-link-target"><span>Bahasa Melayu</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 – birman" lang="my" hreflang="my" data-title="လော့ဂရစ်သမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birman" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Logarithmus" title="Logarithmus – bas-allemand" lang="nds" hreflang="nds" data-title="Logarithmus" data-language-autonym="Plattdüütsch" data-language-local-name="bas-allemand" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Logaritme" title="Logaritme – néerlandais" lang="nl" hreflang="nl" data-title="Logaritme" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Logaritme" title="Logaritme – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Logaritme" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Logaritme" title="Logaritme – norvégien bokmål" lang="nb" hreflang="nb" data-title="Logaritme" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien 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" title="Logaritme – occitan" lang="oc" hreflang="oc" data-title="Logaritme" data-language-autonym="Occitan" data-language-local-name="occitan" 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-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 – pendjabi" lang="pa" hreflang="pa" data-title="ਲਘੂਗਣਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pendjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logarytm" title="Logarytm – polonais" lang="pl" hreflang="pl" data-title="Logarytm" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</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 – Western Punjabi" lang="pnb" hreflang="pnb" data-title="لاگرتھم" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualité"><a href="https://pt.wikipedia.org/wiki/Logaritmo" title="Logaritmo – portugais" lang="pt" hreflang="pt" data-title="Logaritmo" data-language-autonym="Português" data-language-local-name="portugais" 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 – roumain" lang="ro" hreflang="ro" data-title="Logaritm" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualité"><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 – russe" lang="ru" hreflang="ru" data-title="Логарифм" data-language-autonym="Русский" data-language-local-name="russe" 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 – iakoute" lang="sah" hreflang="sah" data-title="Логарифм" data-language-autonym="Саха тыла" data-language-local-name="iakoute" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Logaritmu" title="Logaritmu – sicilien" lang="scn" hreflang="scn" data-title="Logaritmu" data-language-autonym="Sicilianu" data-language-local-name="sicilien" class="interlanguage-link-target"><span>Sicilianu</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 – serbo-croate" lang="sh" hreflang="sh" data-title="Логаритам" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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 – cingalais" lang="si" hreflang="si" data-title="ලඝු ගණක" data-language-autonym="සිංහල" data-language-local-name="cingalais" 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 – Simple English" lang="en-simple" hreflang="en-simple" data-title="Logarithm" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><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 – slovaque" lang="sk" hreflang="sk" data-title="Logaritmus" data-language-autonym="Slovenčina" data-language-local-name="slovaque" 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 – slovène" lang="sl" hreflang="sl" data-title="Logaritem" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Daraunene" title="Daraunene – shona" lang="sn" hreflang="sn" data-title="Daraunene" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logaritmet" title="Logaritmet – albanais" lang="sq" hreflang="sq" data-title="Logaritmet" data-language-autonym="Shqip" data-language-local-name="albanais" class="interlanguage-link-target"><span>Shqip</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 – serbe" lang="sr" hreflang="sr" data-title="Логаритам" data-language-autonym="Српски / srpski" data-language-local-name="serbe" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Logaritm" title="Logaritm – suédois" lang="sv" hreflang="sv" data-title="Logaritm" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Logi" title="Logi – swahili" lang="sw" hreflang="sw" data-title="Logi" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target"><span>Kiswahili</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 – tamoul" lang="ta" hreflang="ta" data-title="மடக்கை" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</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 – thaï" lang="th" hreflang="th" data-title="ลอการิทึม" data-language-autonym="ไทย" data-language-local-name="thaï" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – tagalog" lang="tl" hreflang="tl" data-title="Logaritmo" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Logaritma" title="Logaritma – turc" lang="tr" hreflang="tr" data-title="Logaritma" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</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 – tatar" lang="tt" hreflang="tt" data-title="Логарифм" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</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 – ukrainien" lang="uk" hreflang="uk" data-title="Логарифм" data-language-autonym="Українська" data-language-local-name="ukrainien" 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 – ourdou" lang="ur" hreflang="ur" data-title="لاگرتھم" data-language-autonym="اردو" data-language-local-name="ourdou" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Logarifm" title="Logarifm – ouzbek" lang="uz" hreflang="uz" data-title="Logarifm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualité"><a href="https://vi.wikipedia.org/wiki/Logarit" title="Logarit – vietnamien" lang="vi" hreflang="vi" data-title="Logarit" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Logaritmo" title="Logaritmo – waray" lang="war" hreflang="war" data-title="Logaritmo" data-language-autonym="Winaray" data-language-local-name="waray" 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 – wu" lang="wuu" hreflang="wuu" data-title="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-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 – yiddish" lang="yi" hreflang="yi" data-title="לאגאריטם" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target"><span>ייִדיש</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 – chinois" lang="zh" hreflang="zh" data-title="对数" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</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 – minnan" lang="nan" hreflang="nan" data-title="Tùi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</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 – cantonais" lang="yue" hreflang="yue" data-title="對數" data-language-autonym="粵語" data-language-local-name="cantonais" class="interlanguage-link-target"><span>粵語</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="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</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="Espaces de noms"> <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/Logarithme" title="Voir le contenu de la page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon 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l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Logo_disambig-homophone.svg/20px-Logo_disambig-homophone.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Logo_disambig-homophone.svg/30px-Logo_disambig-homophone.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Logo_disambig-homophone.svg/40px-Logo_disambig-homophone.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour l’article ayant un titre homophone, voir <a href="/wiki/Loga-Rythme" title="Loga-Rythme">Loga-Rythme</a>. </p> </div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Logarithm_plots.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Logarithm_plots.png/220px-Logarithm_plots.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Logarithm_plots.png/330px-Logarithm_plots.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Logarithm_plots.png/440px-Logarithm_plots.png 2x" data-file-width="1706" data-file-height="1294" /></a><figcaption>Tracés des courbes des fonctions logarithmes en base 2, <a href="/wiki/E_(nombre)" title="E (nombre)"><span class="texhtml">e</span></a> et 10.</figcaption></figure> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, un <b>logarithme</b> est la <a href="/wiki/Fonction_r%C3%A9ciproque" class="mw-redirect" title="Fonction réciproque">fonction réciproque</a> d'une <a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, c'est-à-dire que le logarithme de base <span class="texhtml mvar" style="font-style:italic;">b</span> d'un <a href="/wiki/Nombre_r%C3%A9el" title="Nombre réel">nombre réel</a> strictement positif est la <a href="/wiki/Puissance_d%27un_nombre" title="Puissance d'un nombre">puissance</a> à laquelle il faut élever la base <span class="texhtml mvar" style="font-style:italic;">b</span> pour obtenir ce nombre. </p> <div style="margin:0.5em 2em;"><strong>Exemple :</strong> <div style="padding-left:2em; border-left:1px dotted #999;"> <p>Le logarithme en base dix de 1000 est 3 car 10<sup>3</sup> = 10×10×10 = 1000. </p> </div></div> <p>Dans ce cas, le plus simple, le logarithme est le nombre entier qui compte les répétitions de la base multipliée par elle-même. Dans cette opération, multiplier un nombre par la base équivaut à ajouter 1 à son logarithme. L'<a href="/wiki/Exponentiation" title="Exponentiation">exponentiation</a> généralise cette opération de multiplication par soi-même à des puissances intermédiaires entre les entiers, qu'on exprime en nombres réels. </p> <div style="margin:0.5em 2em;"><strong>Exemple :</strong> <div style="padding-left:2em; border-left:1px dotted #999;"> <p>Le logarithme en base dix de la racine de 10, notée <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 {\sqrt {10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7409b0ddbc1f90280265e7bc95dd20626ebf1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.843ex;" alt="{\displaystyle {\sqrt {10}}}"></span>, est 0,5 car </p><p><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 {\sqrt {10}}^{2}={\sqrt {10}}\times {\sqrt {10}}=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {10}}^{2}={\sqrt {10}}\times {\sqrt {10}}=10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd5d49eb7f6671d60d284b6041c6f8ddef673a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.199ex; height:3.343ex;" alt="{\displaystyle {\sqrt {10}}^{2}={\sqrt {10}}\times {\sqrt {10}}=10}"></span>, donc <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 _{10}{\sqrt {10}})+(\log _{10}{\sqrt {10}})=\log _{10}{10}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\log _{10}{\sqrt {10}})+(\log _{10}{\sqrt {10}})=\log _{10}{10}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2214842dc00b74f6b6f19013b83a47f3183305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.37ex; height:3.009ex;" alt="{\displaystyle (\log _{10}{\sqrt {10}})+(\log _{10}{\sqrt {10}})=\log _{10}{10}=1}"></span> </p> </div></div> <p>Le logarithme de base <span class="texhtml mvar" style="font-style:italic;">b</span> du nombre <span class="texhtml mvar" style="font-style:italic;">x</span> se note <span class="texhtml">log<sub><i>b</i></sub> <i>x</i></span>. Si la base est évidente d'après le contexte, ou si elle n'a pas d'importance, on peut écrire simplement <span class="texhtml">log <i>x</i></span>. Par définition, <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^{\log _{b}x}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{\log _{b}x}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/743feead70714b2186ce987a71837976e168dc67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.824ex; height:2.676ex;" alt="{\displaystyle b^{\log _{b}x}=x}"></span>. </p><p><a href="/wiki/John_Napier" title="John Napier">John Napier</a> a développé les logarithmes au début du <abbr class="abbr" title="17ᵉ siècle"><span class="romain">XVII</span><sup style="font-size:72%">e</sup></abbr> siècle. L'utilité du logarithme pour le calcul vient du fait que la fonction logarithme transforme un produit en somme : <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\cdot y)=\log _{b}x+\log _{b}y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x\cdot y)=\log _{b}x+\log _{b}y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7536d0a3b53e44fd3a7c88e07e50549011d5800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.287ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x\cdot y)=\log _{b}x+\log _{b}y\,}"></span>. Pendant trois siècles, la <a href="/wiki/Table_de_logarithmes" title="Table de logarithmes">table de logarithmes</a> et la <a href="/wiki/R%C3%A8gle_%C3%A0_calcul" title="Règle à calcul">règle à calcul</a>, fondée sur une échelle logarithmique, ont servi pour le calcul, jusqu'à leur remplacement, dans le dernier quart du <abbr class="abbr" title="20ᵉ siècle"><span class="romain">XX</span><sup style="font-size:72%">e</sup></abbr> siècle, par des <a href="/wiki/Calculatrice" title="Calculatrice">calculatrices</a> électroniques. </p><p>Le logarithme permet en outre de présenter sous une forme concise des relations entre nombres d'<a href="/wiki/Ordre_de_grandeur" title="Ordre de grandeur">ordre de grandeur</a> très différents. </p><p>Trois fonctions logarithmes sont d'usage courant : </p> <ul><li>le <a href="/wiki/Logarithme_naturel" class="mw-redirect" title="Logarithme naturel">logarithme népérien</a> (ou <i>naturel</i>) dont la base est le <a href="/wiki/E_(nombre)" title="E (nombre)">nombre <span class="texhtml">e</span></a>, est fondamental en <a href="/wiki/Analyse_math%C3%A9matique" class="mw-redirect" title="Analyse mathématique">analyse mathématique</a> car il est la <a href="/wiki/Primitive" title="Primitive">primitive</a> de la fonction <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\mapsto {\tfrac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\tfrac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aba4b386a9233453fbdf76a72b0e675b17383c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.72ex; height:3.343ex;" alt="{\displaystyle x\mapsto {\tfrac {1}{x}}}"></span> s’annulant en 1 et la fonction réciproque de la <a href="/wiki/Fonction_exponentielle" title="Fonction exponentielle">fonction exponentielle</a> ; il est souvent noté <span class="texhtml">ln</span> sauf en <a href="/wiki/Informatique" title="Informatique">informatique</a> ou en <a href="/wiki/Th%C3%A9orie_des_nombres" title="Théorie des nombres">théorie des nombres</a> où <span class="texhtml">log</span> sans autre précision signifie en général logarithme népérien ;</li> <li>le <a href="/wiki/Logarithme_d%C3%A9cimal" title="Logarithme décimal">logarithme décimal</a>, dont la base est 10, reste le plus communément utilisé pour les calculs dans le domaine <a href="/wiki/Technologie" title="Technologie">technologique</a> ainsi qu'en chimie pour le calcul de <a href="/wiki/Potentiel_hydrog%C3%A8ne" title="Potentiel hydrogène">pH</a> ;</li> <li>le <a href="/wiki/Logarithme_binaire" title="Logarithme binaire">logarithme binaire</a>, dont la base est 2, est utile en <a href="/wiki/Informatique_th%C3%A9orique" title="Informatique théorique">informatique théorique</a> et pour certains calculs appliqués.</li></ul> <p>Le <a href="/wiki/Logarithme_complexe" title="Logarithme complexe">logarithme complexe</a> généralise la notion de logarithme aux <a href="/wiki/Nombre_complexe" title="Nombre complexe">nombres complexes</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=1" title="Modifier la section : Motivation" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=1" title="Modifier le code source de la section : Motivation"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une <a href="/wiki/%C3%89chelle_logarithmique" title="Échelle logarithmique">échelle logarithmique</a> permet de représenter sur un même graphique des nombres dont l'<a href="/wiki/Ordre_de_grandeur" title="Ordre de grandeur">ordre de grandeur</a> est très différent. Les <a href="/wiki/Science_appliqu%C3%A9e" title="Science appliquée">sciences appliquées</a> les utilisent fréquemment dans les formules, comme celles qui évaluent la <a href="/wiki/Complexit%C3%A9" title="Complexité">complexité</a> des <a href="/wiki/Algorithmes" class="mw-redirect" title="Algorithmes">algorithmes</a> ou des <a href="/wiki/Fractale" title="Fractale">fractales</a> et celles qui dénombrent les <a href="/wiki/Nombre_premier" title="Nombre premier">nombres premiers</a>. Ils décrivent les <a href="/wiki/Intervalle_(musique)" title="Intervalle (musique)">intervalles musicaux</a> et selon le modèle de <a href="/wiki/Loi_de_Weber-Fechner" title="Loi de Weber-Fechner">Weber-Fechner</a> s'appliquent généralement en <a href="/wiki/Psychophysique" title="Psychophysique">psychophysique</a>. </p><p>Tout logarithme transforme </p> <ul><li>un <a href="/wiki/Multiplication" title="Multiplication">produit</a> en <a href="/wiki/Addition" title="Addition">somme</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 \log _{b}(x\cdot y)=\log _{b}x+\log _{b}y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x\cdot y)=\log _{b}x+\log _{b}y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7536d0a3b53e44fd3a7c88e07e50549011d5800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.287ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x\cdot y)=\log _{b}x+\log _{b}y\,}"></span></li> <li>un <a href="/wiki/Division" title="Division">quotient</a> en <a href="/wiki/Soustraction" title="Soustraction">différence</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 \log _{b}\left({\frac {x}{y}}\right)=\log _{b}x-\log _{b}y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}\left({\frac {x}{y}}\right)=\log _{b}x-\log _{b}y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65cd1c52e5444fedf71c7767d66c9f79c150ade8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.9ex; height:6.176ex;" alt="{\displaystyle \log _{b}\left({\frac {x}{y}}\right)=\log _{b}x-\log _{b}y\,}"></span></li> <li>une puissance en produit : <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^{y})=y\log _{b}x.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <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> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x^{y})=y\log _{b}x.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1540554df73879e6f067c02a46e7a9c79c658ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.399ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x^{y})=y\log _{b}x.\,}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Historique">Historique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=2" title="Modifier la section : Historique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=2" title="Modifier le code source de la section : Historique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Histoire_des_logarithmes_et_des_exponentielles" title="Histoire des logarithmes et des exponentielles">Histoire des logarithmes et des exponentielles</a>.</div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Mirifici_Logarithmorum_canonis_Descriptio.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Mirifici_Logarithmorum_canonis_Descriptio.jpg/220px-Mirifici_Logarithmorum_canonis_Descriptio.jpg" decoding="async" width="220" height="305" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Mirifici_Logarithmorum_canonis_Descriptio.jpg/330px-Mirifici_Logarithmorum_canonis_Descriptio.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Mirifici_Logarithmorum_canonis_Descriptio.jpg/440px-Mirifici_Logarithmorum_canonis_Descriptio.jpg 2x" data-file-width="2301" data-file-height="3193" /></a><figcaption>Page de garde du livre de John Napier de 1614 : Mirifici Logarithmorum Canonis Descriptio</figcaption></figure> <p>La présentation de correspondances entre suites arithmétiques et suites géométriques avec l'observation qu'une somme dans une suite correspond à un produit dans l'autre est ancienne et on la voit déjà chez <a href="/wiki/Archim%C3%A8de" title="Archimède">Archimède</a> (<abbr class="abbr" title="3ᵉ siècle"><span class="romain">III</span><sup style="font-size:72%">e</sup></abbr> siècle <abbr class="abbr nowrap" title="avant Jésus-Christ">av. J.-C.</abbr>), <a href="/wiki/Nicolas_Chuquet" title="Nicolas Chuquet">Chuquet</a> (<abbr class="abbr" title="15ᵉ siècle"><span class="romain">XV</span><sup style="font-size:72%">e</sup></abbr> siècle) et <a href="/wiki/Michael_Stifel" title="Michael Stifel">Stifel</a> (début du <abbr class="abbr" title="16ᵉ siècle"><span class="romain">XVI</span><sup style="font-size:72%">e</sup></abbr> siècle) en Europe<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>, <a href="/wiki/Ibn_Yahy%C4%81_al-Maghrib%C4%AB_al-Samaw%27al" title="Ibn Yahyā al-Maghribī al-Samaw'al">al-Samaw'al</a><sup id="cite_ref-Kouteynoff200620_2-0" class="reference"><a href="#cite_note-Kouteynoff200620-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup> (<abbr class="abbr" title="12ᵉ siècle"><span class="romain">XII</span><sup style="font-size:72%">e</sup></abbr> siècle) et <a href="/wiki/Ibn_Hamza_al-Maghribi" title="Ibn Hamza al-Maghribi">Ibn Hamza al-Maghribi</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup> (fin du <abbr class="abbr" title="16ᵉ siècle"><span class="romain">XVI</span><sup style="font-size:72%">e</sup></abbr> siècle) dans le monde arabe , mais l'observation est plutôt tournée vers une utilisation algébrique<sup id="cite_ref-Kouteynikoff200611_4-0" class="reference"><a href="#cite_note-Kouteynikoff200611-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup>. </p><p>Vers la fin du <abbr class="abbr" title="16ᵉ siècle"><span class="romain">XVI</span><sup style="font-size:72%">e</sup></abbr> siècle, le développement de l'<a href="/wiki/Astronomie" title="Astronomie">astronomie</a> et de la <a href="/wiki/Navigation_maritime" title="Navigation maritime">navigation maritime</a> d'une part et les calculs bancaires d'<a href="/wiki/Int%C3%A9r%C3%AAts_compos%C3%A9s" title="Intérêts composés">intérêts composés</a> d'autre part poussent les mathématiciens à chercher des méthodes de simplification de calculs et en particulier le remplacement des multiplications par des sommes<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup>. L'invention de tables dites logarithmique permettant de faciliter les calculs comportant des produits est l’œuvre de mathématiciens du début du <abbr class="abbr" title="17ᵉ siècle"><span class="romain">XVII</span><sup style="font-size:72%">e</sup></abbr> siècle: <a href="/wiki/Jost_B%C3%BCrgi" title="Jost Bürgi">Jost Bürgi</a><sup id="cite_ref-pedm_6-0" class="reference"><a href="#cite_note-pedm-6"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>, <a href="/wiki/John_Napier" title="John Napier">Neper</a> et <a href="/wiki/Henry_Briggs" title="Henry Briggs">Briggs</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup>, travail poursuivi par <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup>, <a href="/wiki/Ezechiel_de_Decker" title="Ezechiel de Decker">Ezechiel de Decker</a> et <a href="/wiki/Adriaan_Vlacq" title="Adriaan Vlacq">Adriaan Vlacq</a><sup id="cite_ref-pedm_6-1" class="reference"><a href="#cite_note-pedm-6"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>. </p><p>En 1647, <a href="/wiki/Gr%C3%A9goire_de_Saint-Vincent" title="Grégoire de Saint-Vincent">Grégoire de Saint-Vincent</a>, travaillant sur la quadrature de l’<a href="/wiki/Hyperbole_(math%C3%A9matiques)" title="Hyperbole (mathématiques)">hyperbole</a>, définit la fonction primitive de la fonction <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\mapsto {\tfrac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\tfrac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aba4b386a9233453fbdf76a72b0e675b17383c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.72ex; height:3.343ex;" alt="{\displaystyle x\mapsto {\tfrac {1}{x}}}"></span> s’annulant en 1. <a href="/wiki/Christian_Huygens" title="Christian Huygens">Huygens</a> remarquera en 1661 que cette fonction se trouve être une fonction logarithme particulière : le <a href="/wiki/Logarithme_naturel" class="mw-redirect" title="Logarithme naturel">logarithme naturel</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup>. </p><p>La correspondance entre les fonctions exponentielles et logarithmes n’apparaît qu'après le travail de <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> sur la notion de <a href="/wiki/Fonction_(math%C3%A9matiques)" title="Fonction (mathématiques)">fonction</a>, en 1697, et se développe au cours du <a href="/wiki/XVIIIe_si%C3%A8cle" title="XVIIIe siècle"><abbr class="abbr" title="18ᵉ siècle"><span class="romain">XVIII</span><sup style="font-size:72%">e</sup></abbr> siècle</a> dans les écrits d'<a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a><sup id="cite_ref-Barbin20067_10-0" class="reference"><a href="#cite_note-Barbin20067-10"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup>. </p><p>La tentative d'application de la <a href="/wiki/Logarithme_complexe" title="Logarithme complexe">fonction logarithmique à la variable complexe</a> date du <abbr class="abbr" title="18ᵉ siècle"><span class="romain">XVIII</span><sup style="font-size:72%">e</sup></abbr> siècle et donne lieu à une controverse entre <a href="/wiki/Jean_Bernoulli" title="Jean Bernoulli">Bernoulli</a> et <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> résolue par Euler<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Propriétés_des_fonctions_logarithmes"><span id="Propri.C3.A9t.C3.A9s_des_fonctions_logarithmes"></span>Propriétés des fonctions logarithmes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=3" title="Modifier la section : Propriétés des fonctions logarithmes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=3" title="Modifier le code source de la section : Propriétés des fonctions logarithmes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans cette section, nous donnons des propriétés d'une fonction logarithme, quelle que soit sa base <span class="texhtml mvar" style="font-style:italic;">b</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Propriétés_algébriques"><span id="Propri.C3.A9t.C3.A9s_alg.C3.A9briques"></span>Propriétés algébriques</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=4" title="Modifier la section : Propriétés algébriques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=4" title="Modifier le code source de la section : Propriétés algébriques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Identit%C3%A9s_logarithmiques" title="Identités logarithmiques">Identités logarithmiques</a>.</div></div> <p>Les fonctions logarithme sont par définition les <a href="/wiki/Morphisme_de_groupes" title="Morphisme de groupes">morphismes</a> <a href="/wiki/Continuit%C3%A9_(math%C3%A9matiques)" title="Continuité (mathématiques)">continus</a> non <a href="/wiki/Fonction_nulle" title="Fonction nulle">constamment nuls</a> de <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} _{+}^{*},\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} _{+}^{*},\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c438f4e9fa664c88a2acfe45d1bb48ac15d8145" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.84ex; height:3.009ex;" alt="{\displaystyle (\mathbb {R} _{+}^{*},\times )}"></span> vers <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"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </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/5b33b2c9358cbd7bad20aa0b18651d3bba582c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,+)}"></span>. </p><p>Pour tout réel <span class="texhtml mvar" style="font-style:italic;">b</span> strictement positif et différent de 1, le logarithme de base <span class="texhtml mvar" style="font-style:italic;">b</span> : <span class="texhtml">log<sub><i>b</i></sub></span> est la fonction continue définie sur <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"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </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/ef42e5e064679de6752f88a8a2ab8f1e1b6185b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.189ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} _{+}^{*}}"></span> vérifiant l'<a href="/wiki/%C3%89quation_fonctionnelle" title="Équation fonctionnelle">équation fonctionnelle</a> : </p> <dl><dd>pour tous <span class="texhtml mvar" style="font-style:italic;">x</span> et <span class="texhtml mvar" style="font-style:italic;">y</span> réels strictement positifs,</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 _{b}(xy)=\log _{b}(x)+\log _{b}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72b4b7ba4c487ba5c15587d2eff610355605901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.065ex; height:2.843ex;" alt="{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}"></span></dd></dl> <p>et </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 _{b}(b)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(b)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3a58a8d06818394825efc588fa84970424b75f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.977ex; height:2.843ex;" alt="{\displaystyle \log _{b}(b)=1}"></span></dd></dl> <p>Cette définition permet de déduire rapidement les propriétés suivantes : </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 _{b}(1)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(1)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901f6efd3f7b26aa95b855e884a8c2c620ef1fe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.142ex; height:2.843ex;" alt="{\displaystyle \log _{b}(1)=0}"></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 _{b}(x/y)=\log _{b}(x)-\log _{b}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x/y)=\log _{b}(x)-\log _{b}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b760df3fe4794a5497f2a573c1940ee4727c8d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.228ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x/y)=\log _{b}(x)-\log _{b}(y)}"></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 _{b}(x^{y})=y\log _{b}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <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> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x^{y})=y\log _{b}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ba26d51e494fe01d74e0e39a55404cc852ee08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.787ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x^{y})=y\log _{b}(x)}"></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 _{b}(b^{n})=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(b^{n})=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ece2cb3acff9e97e175ca6ccc9f694be8d32a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.428ex; height:2.843ex;" alt="{\displaystyle \log _{b}(b^{n})=n}"></span> pour tout entier naturel <span class="texhtml mvar" style="font-style:italic;">n</span>, puis pour tout entier relatif <span class="texhtml mvar" style="font-style:italic;">n</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 _{b}(b^{r})=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(b^{r})=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfdf5e80986c7298d2af7cb1b3b44e5db5a37dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.837ex; height:2.843ex;" alt="{\displaystyle \log _{b}(b^{r})=r}"></span> pour tout rationnel <span class="texhtml mvar" style="font-style:italic;">r</span>.</dd></dl> <p>Comme tout réel strictement positif <span class="texhtml mvar" style="font-style:italic;">x</span> est la <a href="/wiki/Limite_d%27une_suite" title="Limite d'une suite">limite d'une suite</a> dont le terme général est de la forme <span class="texhtml mvar" style="font-style:italic;">b<sup>r<sub>n</sub></sup></span>, où <span class="texhtml">(<i>r<sub>n</sub></i>)</span> est une suite de rationnels convergeant vers un réel <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span>, on détermine <span class="texhtml">log<sub><i>b</i></sub>(<i>x</i>)</span> comme étant la limite de <span class="texhtml mvar" style="font-style:italic;">r<sub>n</sub></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Changement_de_base">Changement de base</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=5" title="Modifier la section : Changement de base" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=5" title="Modifier le code source de la section : Changement de base"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Deux fonctions logarithmes ne diffèrent que d’une constante multiplicative : pour tous réels strictement positifs <span class="texhtml mvar" style="font-style:italic;">a</span> et <span class="texhtml mvar" style="font-style:italic;">b</span> différents de 1 et pour tout réel <span class="texhtml"><i>x</i> > 0</span>, </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 _{b}(x)={\frac {\log _{a}(x)}{\log _{a}(b)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x)={\frac {\log _{a}(x)}{\log _{a}(b)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2d1396dcd579a41d083caa7d5e221889337526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.196ex; height:6.509ex;" alt="{\displaystyle \log _{b}(x)={\frac {\log _{a}(x)}{\log _{a}(b)}}}"></span>.</dd></dl> <p>Toutes les fonctions logarithmes peuvent donc s’exprimer à l’aide d’une seule, par exemple la fonction logarithme népérien : pour tout réel strictement positif <span class="texhtml mvar" style="font-style:italic;">b</span> différent de 1 et pour tout réel <span class="texhtml"><i>x</i> > 0</span>, </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 _{b}(x)={\frac {\ln(x)}{\ln(b)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x)={\frac {\ln(x)}{\ln(b)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52a93d813551fae4ee9a530804b1ca385e094c01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.061ex; height:6.509ex;" alt="{\displaystyle \log _{b}(x)={\frac {\ln(x)}{\ln(b)}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Dérivée"><span id="D.C3.A9riv.C3.A9e"></span>Dérivée</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=6" title="Modifier la section : Dérivée" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=6" title="Modifier le code source de la section : Dérivée"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La fonction <span class="texhtml">log<sub><i>b</i></sub></span> est <a href="/wiki/D%C3%A9riv%C3%A9e" title="Dérivée">dérivable</a> sur <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"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </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/ef42e5e064679de6752f88a8a2ab8f1e1b6185b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.189ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} _{+}^{*}}"></span> de dérivée : </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 _{b}'(x)={\frac {1}{x\ln(b)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>′</mo> </msubsup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}'(x)={\frac {1}{x\ln(b)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbda63e2ee151f62ebaeb0050a0d7bf088d1ba59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.446ex; height:6.009ex;" alt="{\displaystyle \log _{b}'(x)={\frac {1}{x\ln(b)}}}"></span> qui a même signe que <span class="texhtml">ln(<i>b</i>)</span>.</dd></dl> <p>Donc la fonction <span class="texhtml">log<sub><i>b</i></sub></span> est strictement monotone, croissante quand <span class="texhtml mvar" style="font-style:italic;">b</span> est supérieur à 1, décroissante dans le cas contraire. </p> <div class="mw-heading mw-heading3"><h3 id="Nombre_de_chiffres_avant_la_virgule">Nombre de chiffres avant la virgule</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=7" title="Modifier la section : Nombre de chiffres avant la virgule" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=7" title="Modifier le code source de la section : Nombre de chiffres avant la virgule"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si <span class="texhtml mvar" style="font-style:italic;">b</span> est un entier supérieur ou égal à 2 et <span class="texhtml"><i>x</i> > 0</span>, la <a href="/wiki/Base_(arithm%C3%A9tique)#Développement_en_base_entière" title="Base (arithmétique)">représentation propre de <span class="texhtml mvar" style="font-style:italic;">x</span> en base</a> <i><span class="texhtml mvar" style="font-style:italic;">b</span></i> possède <span class="texhtml mvar" style="font-style:italic;">n</span> chiffres avant la virgule si et seulement si <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-1}\leqslant x<b^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⩽</mo> <mi>x</mi> <mo><</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{n-1}\leqslant x<b^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48b1b148953961b9ebbaf06ac850ff7702ab9aa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.059ex; height:2.843ex;" alt="{\displaystyle b^{n-1}\leqslant x<b^{n}}"></span>, soit <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 n-1\leqslant \log _{b}x<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>⩽</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1\leqslant \log _{b}x<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd33b395f8c2a9294ffeb49f60ffbff596cffdcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.615ex; height:2.676ex;" alt="{\displaystyle n-1\leqslant \log _{b}x<n}"></span>. Le nombre de chiffres <span class="texhtml mvar" style="font-style:italic;">n</span> est donc égal à <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 \left\lfloor {\log _{b}x}\right\rfloor +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⌊</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> <mo>⌋</mo> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\lfloor {\log _{b}x}\right\rfloor +1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee8bb8787724fd620844c0c76486ee8aa7cdb49f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.694ex; height:2.843ex;" alt="{\displaystyle \left\lfloor {\log _{b}x}\right\rfloor +1}"></span>. </p><p>Et lorsque <span class="texhtml mvar" style="font-style:italic;">x</span> tend vers l'infini, on a donc <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\sim n(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>∼</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}x\sim n(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5040ee932b39b413bd47ac8eaf40f9dc19cda795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.258ex; height:2.843ex;" alt="{\displaystyle \log _{b}x\sim n(x)}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Fonction_réciproque_(antilogarithme)"><span id="Fonction_r.C3.A9ciproque_.28antilogarithme.29"></span>Fonction réciproque (<a href="/wiki/Antilogarithme" title="Antilogarithme">antilogarithme</a>)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=8" title="Modifier la section : Fonction réciproque (antilogarithme)" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=8" title="Modifier le code source de la section : Fonction réciproque (antilogarithme)"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Exponentielle_de_base_a" title="Exponentielle de base a">Exponentielle de base <span class="texhtml mvar" style="font-style:italic;">b</span></a>.</div></div> <p><span id="Fonction_inverse"></span> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Logarithm_inversefunctiontoexp.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Logarithm_inversefunctiontoexp.svg/220px-Logarithm_inversefunctiontoexp.svg.png" decoding="async" width="220" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Logarithm_inversefunctiontoexp.svg/330px-Logarithm_inversefunctiontoexp.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Logarithm_inversefunctiontoexp.svg/440px-Logarithm_inversefunctiontoexp.svg.png 2x" data-file-width="240" data-file-height="279" /></a><figcaption>Représentation dans le cas <span class="texhtml"><i>b</i> > 1</span>. Le graphe de la fonction logarithmique <span class="texhtml">log<sub><i>b</i></sub>(<i>x</i>)</span> (bleu) est obtenu en <a href="/wiki/R%C3%A9flexion_(math%C3%A9matiques)" title="Réflexion (mathématiques)">reflétant</a> celui de la fonction <span class="texhtml mvar" style="font-style:italic;">b<sup>x</sup></span> (rouge) par rapport à la diagonale <span class="nowrap"><i>x = y.</i></span></figcaption></figure> <p>La fonction <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}:\mathbb {R} _{+}^{*}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>:</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}:\mathbb {R} _{+}^{*}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07209bb023d0635e6343f622a9a0f263c9d1574a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.328ex; height:3.009ex;" alt="{\displaystyle \log _{b}:\mathbb {R} _{+}^{*}\to \mathbb {R} }"></span> est la <a href="/wiki/Bijection" title="Bijection">bijection</a> <a href="/wiki/Bijection_r%C3%A9ciproque" title="Bijection réciproque">réciproque</a> de la fonction exponentielle de base <span class="texhtml mvar" style="font-style:italic;">b</span><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup>, parfois appelée antilogarithme de base <span class="texhtml mvar" style="font-style:italic;">b</span> : </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 \operatorname {antilog_{b}} :\mathbb {R} \to \mathbb {R} _{+}^{*},\;x\mapsto b^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <msub> <mi mathvariant="normal">g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </msub> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>,</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo stretchy="false">↦</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {antilog_{b}} :\mathbb {R} \to \mathbb {R} _{+}^{*},\;x\mapsto b^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54d3acd11206551f915c1d0f6a2eab4a5987a43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.336ex; height:3.009ex;" alt="{\displaystyle \operatorname {antilog_{b}} :\mathbb {R} \to \mathbb {R} _{+}^{*},\;x\mapsto b^{x}}"></span>.</dd></dl> <p>Autrement dit, les deux façons possibles de combiner (ou <a href="/wiki/Composition_de_fonctions" title="Composition de fonctions">composer</a>) les logarithmes et l’élévation à des puissances redonnent le nombre original : </p> <ul><li>pour tout réel <span class="texhtml"><i>x</i></span>, prendre la puissance <span class="nowrap"><span class="texhtml"><i>x</i></span>-ième</span> de <span class="texhtml"><i>b</i></span>, puis le logarithme en base <span class="texhtml"><i>b</i></span> de cette puissance, redonne <span class="texhtml"><i>x</i></span> :<center><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 \forall x\in \mathbb {R} _{+}^{*}\quad \log _{b}(b^{x})=x\log _{b}(b)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mspace width="1em" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in \mathbb {R} _{+}^{*}\quad \log _{b}(b^{x})=x\log _{b}(b)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1909a8c402f18b46a20890d71137c94d4970eed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.21ex; height:3.009ex;" alt="{\displaystyle \forall x\in \mathbb {R} _{+}^{*}\quad \log _{b}(b^{x})=x\log _{b}(b)=x}"></span> ;</center></li> <li>inversement, pour tout réel <span class="texhtml"><i>y</i></span> strictement positif, prendre d'abord le logarithme en base <span class="texhtml"><i>b</i></span>, puis élever <span class="texhtml"><i>b</i></span> à sa puissance, redonne <span class="texhtml"><i>y</i></span> :<center><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^{\log _{b}(y)}=y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{\log _{b}(y)}=y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dee1975e870ed438a4fc27318eda5220dc335b0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.065ex; height:3.176ex;" alt="{\displaystyle b^{\log _{b}(y)}=y.}"></span></center></li></ul> <p>Les fonctions réciproques sont étroitement liées aux fonctions originales. Leurs <a href="/wiki/Graphe_d%27une_fonction" title="Graphe d'une fonction">graphes</a>, qui se correspondent lorsqu’on échange les coordonnées <span class="texhtml"><i>x</i></span> et <span class="texhtml"><i>y</i></span> (ou par réflexion par rapport à la diagonale <span class="texhtml"><i>x</i> = <i>y</i></span>), sont montrés à droite dans le cas où <span class="texhtml"><i>b</i></span> est un réel strictement supérieur à 1 : un point <span class="texhtml">(<i>u</i>, <i>t</i> = <i>b<sup>u</sup></i>)</span> sur le graphe (rouge) de la fonction antilogarithme <span class="texhtml"><i>x</i> ↦ <i>b<sup>x</sup></i></span> fournit un point <span class="texhtml">(<i>t</i>, <i>u</i> = log<sub><i>b</i></sub>(<i>t</i>))</span> sur le graphe (bleu) du logarithme et vice versa. Comme <span class="texhtml"><i>b</i> > 1</span>, la fonction <span class="texhtml">log<sub><i>b</i></sub></span> est <a href="/wiki/Fonction_monotone" title="Fonction monotone">croissante</a> et quand <span class="texhtml"><i>x</i></span> tend vers <span class="texhtml">+∞</span>, <span class="texhtml">log<sub><i>b</i></sub>(<i>x</i>)</span> <a href="/wiki/Limite_(math%C3%A9matiques)" title="Limite (mathématiques)">tend vers</a> <span class="texhtml">+∞</span>, tandis que lorsque <span class="texhtml"><i>x</i></span> approche zéro, <span class="texhtml">log<sub><i>b</i></sub>(<i>x</i>)</span> tend vers <span class="texhtml">–∞</span>. Dans le cas où le réel <span class="texhtml"><i>b</i></span> est strictement compris entre 0 et 1, la fonction <span class="texhtml">log<sub><i>b</i></sub></span> est décroissante et ces limites sont interverties. </p><p>En matière de calcul, l'antilog ramène des logarithmes aux valeurs. Soit à évaluer une formule <span class="texhtml mvar" style="font-style:italic;">F</span> combinant multiplications, divisions et exponentiations, et soit <span class="texhtml mvar" style="font-style:italic;">f</span> la formule définissant le logarithme de <span class="texhtml mvar" style="font-style:italic;">F</span> en combinant sommes, différences et produits des (logarithmes) des données. La valeur de <span class="texhtml mvar" style="font-style:italic;">F</span> peut s'obtenir comme l'antilog de la valeur de <span class="texhtml mvar" style="font-style:italic;">f</span>, ce qui conclut le calcul. On peut ainsi remplacer l'évaluation <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\times y\times z)^{1/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> <mo>×</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=(x\times y\times z)^{1/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51129b961cffd73fe831f5f7529d744fd74a6c47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.601ex; height:3.343ex;" alt="{\displaystyle F=(x\times y\times z)^{1/3}}"></span> </p><p> par </p><center><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=\operatorname {antilog} _{b}\left({\frac {\log _{b}(x)+\log _{b}(y)+\log _{b}(z)}{3}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <msub> <mi>antilog</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\operatorname {antilog} _{b}\left({\frac {\log _{b}(x)+\log _{b}(y)+\log _{b}(z)}{3}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f9bb943358f66c5295a498d868ea8eda728265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.423ex; height:6.343ex;" alt="{\displaystyle F=\operatorname {antilog} _{b}\left({\frac {\log _{b}(x)+\log _{b}(y)+\log _{b}(z)}{3}}\right)}"></span>.</center> <div class="mw-heading mw-heading2"><h2 id="Fonctions_logarithme_courantes">Fonctions logarithme courantes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=9" title="Modifier la section : Fonctions logarithme courantes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=9" title="Modifier le code source de la section : Fonctions logarithme courantes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Logarithme_népérien"><span id="Logarithme_n.C3.A9p.C3.A9rien"></span>Logarithme népérien</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=10" title="Modifier la section : Logarithme népérien" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=10" title="Modifier le code source de la section : Logarithme népérien"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Logarithme_n%C3%A9p%C3%A9rien" title="Logarithme népérien">Logarithme népérien</a>.</div></div> <p>Le logarithme népérien, ou logarithme naturel, est la fonction logarithme dont la <a href="/wiki/D%C3%A9riv%C3%A9e" title="Dérivée">dérivée</a> est la <a href="/wiki/Fonction_inverse" title="Fonction inverse">fonction inverse</a> définie de <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"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </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/ef42e5e064679de6752f88a8a2ab8f1e1b6185b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.189ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} _{+}^{*}}"></span> dans <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> : <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\mapsto {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto {\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d207e4d6d9278902dd0e1a54b1dab01f5b5037fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.11ex; height:5.176ex;" alt="{\displaystyle x\mapsto {\frac {1}{x}}}"></span>. </p> <dl><dd>La fonction de Neper est par convention notée « <span class="texhtml">ln</span> »<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite_crochet">[</span>13<span class="cite_crochet">]</span></a></sup> ou « <span class="texhtml">log</span> », notation couramment utilisée en <a href="/wiki/Th%C3%A9orie_des_nombres" title="Théorie des nombres">théorie des nombres</a> et en informatique<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite_crochet">[</span>14<span class="cite_crochet">]</span></a></sup>.</dd></dl> <dl><dd>La base de la fonction logarithme népérien, notée <a href="/wiki/E_(nombre)" title="E (nombre)">e</a>, est appelée nombre de Néper<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite_crochet">[</span>15<span class="cite_crochet">]</span></a></sup> ou nombre d'Euler<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite_crochet">[</span>16<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite_crochet">[</span>17<span class="cite_crochet">]</span></a></sup>.</dd></dl> <p>Une <a href="/wiki/Valeur_approch%C3%A9e" title="Valeur approchée">valeur approchée</a> est : </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 \mathrm {e} \approx 2{,}718}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>≈</mo> <mn>2,718</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} \approx 2{,}718}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c844baeb87bbd5589acbfe876caf5c35f939407e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.428ex; height:2.509ex;" alt="{\displaystyle \mathrm {e} \approx 2{,}718}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Logarithme_décimal"><span id="Logarithme_d.C3.A9cimal"></span>Logarithme décimal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=11" title="Modifier la section : Logarithme décimal" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=11" title="Modifier le code source de la section : Logarithme décimal"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Logarithme_d%C3%A9cimal" title="Logarithme décimal">Logarithme décimal</a>.</div></div> <p>C’est le logarithme le plus pratique dans les calculs numériques manuels, il est noté <span class="texhtml">log</span> ou <span class="texhtml">log<sub>10</sub></span>. La norme ISO 80000-2<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite_crochet">[</span>18<span class="cite_crochet">]</span></a></sup> indique que log<sub>10</sub> devrait être noté <i>lg</i>, mais cette notation est rarement utilisée. </p><p>On le retrouve dans la création des <a href="/wiki/%C3%89chelle_logarithmique" title="Échelle logarithmique">échelles logarithmiques</a>, les <a href="/wiki/Rep%C3%A8re_semi-logarithmique" title="Repère semi-logarithmique">repères semi-logarithmiques</a> ou <a href="/wiki/Rep%C3%A8re_log-log" title="Repère log-log">log-log</a>, dans la <a href="/wiki/R%C3%A8gle_%C3%A0_calcul" title="Règle à calcul">règle à calcul</a>, dans le calcul du <a href="/wiki/Potentiel_hydrog%C3%A8ne" title="Potentiel hydrogène">pH</a>, dans l’unité du <a href="/wiki/D%C3%A9cibel" title="Décibel">décibel</a>. </p><p>Il précise à quelle puissance il faut élever 10 pour retrouver le nombre de départ : l'<a href="/wiki/Image_(math%C3%A9matiques)" title="Image (mathématiques)">image</a> d'un nombre par <span class="texhtml">log</span> est l'<a href="/wiki/Entier_relatif" title="Entier relatif">entier relatif</a> auquel il faut élever 10 pour obtenir l'<a href="/wiki/Ant%C3%A9c%C3%A9dent_(math%C3%A9matiques)" title="Antécédent (mathématiques)">antécédent</a>. Par exemple : </p> <dl><dd>En base dix :</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 _{10}(10)=1{\text{ car }}10^{1}=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> car </mtext> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(10)=1{\text{ car }}10^{1}=10}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7dee6fbc2e98052ff964200e913caac2ba003c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.313ex; height:3.176ex;" alt="{\displaystyle \log _{10}(10)=1{\text{ car }}10^{1}=10}"></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 _{10}(100)=2{\text{ car }}10^{2}=100}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>100</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> car </mtext> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>100</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(100)=2{\text{ car }}10^{2}=100}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b10b46514c14008b5616453943105080daefbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.638ex; height:3.176ex;" alt="{\displaystyle \log _{10}(100)=2{\text{ car }}10^{2}=100}"></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 _{10}(1000)=3{\text{ car }}10^{3}=1000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1000</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> car </mtext> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>1000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(1000)=3{\text{ car }}10^{3}=1000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9194d98c9dce5708e6579f1a2ccfe779c034aaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.963ex; height:3.176ex;" alt="{\displaystyle \log _{10}(1000)=3{\text{ car }}10^{3}=1000}"></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 _{10}(0,01)=-2{\text{ car }}10^{-2}=0,01}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>01</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> car </mtext> </mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>01</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(0,01)=-2{\text{ car }}10^{-2}=0,01}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aeed3f9bff30d46b6b1c41953d8227da43f5830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.793ex; height:3.176ex;" alt="{\displaystyle \log _{10}(0,01)=-2{\text{ car }}10^{-2}=0,01}"></span></dd></dl> <p>La valeur du logarithme d’autres nombres que des puissances de 10 demande un calcul approché. Le calcul de <span class="texhtml">log(2)</span> par exemple peut se faire à la main, en remarquant que 2<sup>10</sup> ≈ 1000 donc <span class="texhtml">10 log<sub>10</sub>(2) ≈ 3</span> donc <span class="texhtml">log<sub>10</sub>(2) ≈ 0,3</span>. </p><p>Pour tout réel strictement positif <span class="texhtml mvar" style="font-style:italic;">b</span> différent de 1 et pour tout réel <span class="texhtml"><i>x</i> > 0</span>, </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 _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <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> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/148103ad3a4cfff9c7d030a3352dc9363aab3617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.97ex; height:6.509ex;" alt="{\displaystyle \log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Logarithme_binaire">Logarithme binaire</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=12" title="Modifier la section : Logarithme binaire" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=12" title="Modifier le code source de la section : Logarithme binaire"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Logarithme_binaire" title="Logarithme binaire">Logarithme binaire</a>.</div></div> <p>La norme ISO 80 000 recommande de noter <span class="texhtml">lb</span> le logarithme en base 2<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite_crochet">[</span>19<span class="cite_crochet">]</span></a></sup>. </p><p>Le <a href="/wiki/Logarithme_binaire" title="Logarithme binaire">logarithme binaire</a>, d'usage spécialisé dans le calcul des <a href="/wiki/Intervalle_(musique)" title="Intervalle (musique)">intervalles musicaux</a> à partir d'un rapport de <a href="/wiki/Fr%C3%A9quence" title="Fréquence">fréquences</a>, pour obtenir des <a href="/wiki/Octave_(musique)" title="Octave (musique)">octaves</a>, des <a href="/wiki/Demi-ton" title="Demi-ton">demi-tons</a> ou des <a href="/wiki/Cent_et_savart" class="mw-redirect" title="Cent et savart">cents</a>, a trouvé beaucoup plus d'application en <a href="/wiki/Informatique" title="Informatique">informatique</a>. Les <a href="/wiki/Ordinateur" title="Ordinateur">ordinateurs</a> travaillant en <a href="/wiki/Syst%C3%A8me_binaire" title="Système binaire">système binaire</a>, le calcul d'un logarithme en base 2 se fait par l'algorithme le plus précis et le plus efficace. </p><p>Un nombre <i>x</i> codé en <a href="/wiki/Virgule_flottante" title="Virgule flottante">virgule flottante</a> binaire se décompose en une <a href="/wiki/Mantisse" title="Mantisse">mantisse</a> <i>m</i>, comprise entre 1 (inclus) et 2 (exclu) et un <a href="/wiki/Exposant_(math%C3%A9matiques)" title="Exposant (mathématiques)">exposant</a> <i>p</i>, indiquant la <a href="/wiki/Puissance_de_deux" title="Puissance de deux">puissance de 2</a> qui multiplie la mantisse pour obtenir le nombre. L'exposant est la <a href="/wiki/Partie_enti%C3%A8re_et_partie_fractionnaire" title="Partie entière et partie fractionnaire">partie entière</a> du logarithme binaire, tandis que le logarithme binaire de la mantisse est compris entre 0 (inclus) et 1 (exclu). </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 x=2^{p}\times m\Longrightarrow {\textrm {lb}}(x)=p+{\textrm {lb}}(m).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>×</mo> <mi>m</mi> <mo stretchy="false">⟹</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>lb</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>lb</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2^{p}\times m\Longrightarrow {\textrm {lb}}(x)=p+{\textrm {lb}}(m).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798e9a08d212aac76f98a78cbe111fa64654f2db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.249ex; height:2.843ex;" alt="{\displaystyle x=2^{p}\times m\Longrightarrow {\textrm {lb}}(x)=p+{\textrm {lb}}(m).}"></span></dd></dl> <p>Ce qui ramène le calcul à celui du logarithme binaire d'un nombre entre 1 (inclus) et 2 (exclu). Si on multiplie ce nombre par lui-même, et que le résultat dépasse 2, c'est que le nombre est supérieur à <span class="racine">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> : le chiffre suivant, après la virgule, est un 1, dans le cas contraire, c'est un 0. On continue par <a href="/wiki/It%C3%A9ration" title="Itération">itération</a> jusqu'à la précision souhaitée. </p><p>Les deux logarithmes précédents se déduisent de celui-ci par : </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 \ln(x)={\frac {\mathrm {lb} (x)}{\mathrm {lb} (\mathrm {e} )}}{\text{ et }}\log _{10}(x)={\frac {\mathrm {lb} (x)}{\mathrm {lb} (10)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">b</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> et </mtext> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">b</mi> </mrow> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)={\frac {\mathrm {lb} (x)}{\mathrm {lb} (\mathrm {e} )}}{\text{ et }}\log _{10}(x)={\frac {\mathrm {lb} (x)}{\mathrm {lb} (10)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78491bf15a7a9cdfbfeaade0dba8fb069060c890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.572ex; height:6.509ex;" alt="{\displaystyle \ln(x)={\frac {\mathrm {lb} (x)}{\mathrm {lb} (\mathrm {e} )}}{\text{ et }}\log _{10}(x)={\frac {\mathrm {lb} (x)}{\mathrm {lb} (10)}}}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Cologarithme">Cologarithme</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=13" title="Modifier la section : Cologarithme" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=13" title="Modifier le code source de la section : Cologarithme"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Cologarithme" title="Cologarithme">Cologarithme</a>.</div></div> <p>Le <a href="/wiki/Cologarithme" title="Cologarithme">cologarithme</a> d'un nombre est l'opposé du logarithme de ce nombre et le logarithme de son inverse<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite_crochet">[</span>20<span class="cite_crochet">]</span></a></sup> : <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 {colog} _{b}x=-\log _{b}x=\log _{b}{\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>colog</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {colog} _{b}x=-\log _{b}x=\log _{b}{\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793823f9a8aeab235f52aa39ca84e68d26a24429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.302ex; height:5.176ex;" alt="{\displaystyle \operatorname {colog} _{b}x=-\log _{b}x=\log _{b}{\frac {1}{x}}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Généralisations"><span id="G.C3.A9n.C3.A9ralisations"></span>Généralisations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=14" title="Modifier la section : Généralisations" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=14" title="Modifier le code source de la section : Généralisations"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le <a href="/wiki/Logarithme_complexe" title="Logarithme complexe">logarithme complexe</a> est la fonction réciproque de l'<a href="/wiki/Exponentielle_complexe" title="Exponentielle complexe">exponentielle complexe</a> et généralise ainsi la notion de logarithme aux <a href="/wiki/Nombre_complexe" title="Nombre complexe">nombres complexes</a>. Le <a href="/wiki/Logarithme_discret" title="Logarithme discret">logarithme discret</a> généralise les logarithmes aux <a href="/wiki/Groupe_cyclique" title="Groupe cyclique">groupes cycliques</a> et a des applications en <a href="/wiki/Cryptographie_%C3%A0_cl%C3%A9_publique" class="mw-redirect" title="Cryptographie à clé publique">cryptographie à clé publique</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=15" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=15" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé <span class="plainlinks">« <a class="external text" href="https://en.wikipedia.org/wiki/Logarithm?oldid=408909865">Logarithm</a> » <small>(<a class="external text" href="https://en.wikipedia.org/wiki/Logarithm?action=history">voir la liste des auteurs</a>)</small></span>.</li></ul> <div class="references-small decimal" style=""><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Kouteynikoff2006"><span class="ouvrage" id="Odile_Kouteynikoff2006">Odile Kouteynikoff, <cite style="font-style:normal">« Invention de nombres : calculs ou résolutions »</cite>, dans Commissionn inter-Irem d'Épistémologie et d'histoire des mathématiques, <cite class="italique">Histoire de logarithmes</cite>, Ellipses, <time>2006</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">11-38</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Histoire+de+logarithmes&rft.atitle=Invention+de+nombres+%3A+calculs+ou+r%C3%A9solutions&rft.pub=Ellipses&rft.aulast=Kouteynikoff&rft.aufirst=Odile&rft.date=2006&rft.pages=11-38&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogarithme"></span></span></span>, p. 11</span> </li> <li id="cite_note-Kouteynoff200620-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-Kouteynoff200620_2-0">↑</a> </span><span class="reference-text"><a href="#Kouteynoff2006">Kouteynoff 2006</a>, <abbr class="abbr" title="page(s)">p.</abbr> 20. </span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Ageron"><span class="ouvrage" id="Pierre_Ageron">Pierre Ageron, « <a rel="nofollow" class="external text" href="https://ageron.users.lmno.cnrs.fr/34%20-%20Actes%20-%20Section%20II-1%20-%2016%20Pierre%20Ageron%20-%20339-359#:~:text=Ibn%20Hamza%20s'intéressait%20aux,de%20la%20science%20des%20logarithmes."><cite style="font-style:normal;">Ibn Hamza a-t-il découvert les logarithmes ? Constitution et circulation du discours islamocentré sur l’histoire des mathématiques</cite></a> » <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>, sur <span class="italique">IREM de Basse-Normandie & Université de Caen</span></span></span></span> </li> <li id="cite_note-Kouteynikoff200611-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-Kouteynikoff200611_4-0">↑</a> </span><span class="reference-text"><a href="#Kouteynikoff2006">Kouteynikoff 2006</a>, <abbr class="abbr" title="page(s)">p.</abbr> 11. </span> </li> <li id="cite_note-5"><span class="mw-cite-backlink noprint"><a href="#cite_ref-5">↑</a> </span><span class="reference-text">Jean-Pierre Friedelmeyer, <a rel="nofollow" class="external text" href="https://lewebpedagogique.com/h4mathsts1/files/2013/12/105_122_AM61-4-1.pdf">L'invention des logarithmes par Neper et le calcul des logarithmes décimaux par Briggs</a>.</span> </li> <li id="cite_note-pedm-6"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-pedm_6-0">a</a> et <a href="#cite_ref-pedm_6-1">b</a></sup> </span><span class="reference-text"><i>Petite encyclopédie de mathématiques</i>, <a href="/wiki/Didier_(maison_d%27%C3%A9dition)" title="Didier (maison d'édition)">Didier</a>, 1980, <abbr class="abbr" title="page">p.</abbr> 72</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink noprint"><a href="#cite_ref-7">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Barbin2006"><span class="ouvrage" id="Évelyne_Barbin2006">Évelyne Barbin, <cite style="font-style:normal">« Présentation: pour une approche historique des logarithmes et des exponentielles »</cite>, dans Commissionn inter-Irem d'Épistémologie et d'histoire des mathématiques, <cite class="italique">Histoire de logarithmes</cite>, Ellipses, <time>2006</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">5-10</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Histoire+de+logarithmes&rft.atitle=Pr%C3%A9sentation%3A+pour+une+approche+historique+des+logarithmes+et+des+exponentielles&rft.pub=Ellipses&rft.aulast=Barbin&rft.aufirst=%C3%89velyne&rft.date=2006&rft.pages=5-10&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogarithme"></span></span></span>, p.6</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink noprint"><a href="#cite_ref-8">↑</a> </span><span class="reference-text"><span class="ouvrage">« <a rel="nofollow" class="external text" href="https://www.e-rara.ch/zut/wihibe/content/titleinfo/1405839"><cite style="font-style:normal;">Chilias Logarithmorum</cite></a> », sur <span class="italique">e-rara.ch</span></span>.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink noprint"><a href="#cite_ref-9">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Ferrand,_Laurent_Koelblen,_Matthieu_Romagny2008"><span class="ouvrage" id="Emmanuel_Ferrand,_Laurent_Koelblen,_Matthieu_Romagny2008">Emmanuel Ferrand, Laurent Koelblen, Matthieu Romagny, « <a rel="nofollow" class="external text" href="https://perso.univ-rennes1.fr/matthieu.romagny/capes_0809/histoire.pdf"><cite style="font-style:normal;">Un peu d’histoire</cite></a> », <time class="nowrap" datetime="2008-09-17" data-sort-value="2008-09-17">17 septembre 2008</time></span></span></span> </li> <li id="cite_note-Barbin20067-10"><span class="mw-cite-backlink noprint"><a href="#cite_ref-Barbin20067_10-0">↑</a> </span><span class="reference-text"><a href="#Barbin2006">Barbin 2006</a>, <abbr class="abbr" title="page(s)">p.</abbr> 7. </span> </li> <li id="cite_note-11"><span class="mw-cite-backlink noprint"><a href="#cite_ref-11">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Verley2006"><span class="ouvrage" id="Jean-Luc_Verley2006">Jean-Luc Verley, <cite style="font-style:normal">« La controverse des logarithmes des nombres négatifs et imagianires »</cite>, dans Commissionn inter-Irem d'Épistémologie et d'histoire des mathématiques, <cite class="italique">Histoire de logarithmes</cite>, Ellipses, <time>2006</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">269-288</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Histoire+de+logarithmes&rft.atitle=La+controverse+des+logarithmes+des+nombres+n%C3%A9gatifs+et+imagianires&rft.pub=Ellipses&rft.aulast=Verley&rft.aufirst=Jean-Luc&rft.date=2006&rft.pages=269-288&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogarithme"></span></span></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink noprint"><a href="#cite_ref-12">↑</a> </span><span class="reference-text"><span class="ouvrage" id="James_Stewart_(en)Catégorie:Article_contenant_un_appel_à_traduction_en_anglais2012"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <span class="nom_auteur"><a href="/w/index.php?title=James_Stewart_(math%C3%A9maticien)&action=edit&redlink=1" class="new" title="James Stewart (mathématicien) (page inexistante)">James Stewart</a> <a href="https://en.wikipedia.org/wiki/James_Stewart_(mathematician)" class="extiw" title="en:James Stewart (mathematician)"><span class="indicateur-langue" title="Article en anglais : « James Stewart (mathematician) »">(en)</span></a></span>, <cite class="italique" lang="en">Single Variable Calculus : Early Transcendentals</cite>, Thomson Brooks/Cole, <time>2012</time>, <abbr class="abbr" title="septième">7<sup>e</sup></abbr> <abbr class="abbr" title="édition">éd.</abbr> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="//books.google.com/books?id=h4Auk70bJogC&pg=PA58">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Single+Variable+Calculus&rft.pub=Thomson+Brooks%2FCole&rft.edition=7&rft.stitle=Early+Transcendentals&rft.aulast=James+Stewart&rft.date=2012&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogarithme"></span></span>, section 1.6.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink noprint"><a href="#cite_ref-13">↑</a> </span><span class="reference-text">La norme AFNOR NF X 02-1 01, de 1961, recommande la notation ln (<i>Tables numériques</i> de J. Laborde, 1976, p. VI).</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink noprint"><a href="#cite_ref-14">↑</a> </span><span class="reference-text">Langages <a href="/wiki/C_(langage)" title="C (langage)">C</a>, <a href="/wiki/Java_(technique)" title="Java (technique)">Java</a>, <a href="/wiki/Javascript" class="mw-redirect" title="Javascript">Javascript</a>, <abbr class="abbr" title="et cetera">etc.</abbr></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink noprint"><a href="#cite_ref-15">↑</a> </span><span class="reference-text"><span class="ouvrage" id="GuininJoppin2003"><span class="ouvrage" id="D._GuininB._Joppin2003">D. Guinin et B. Joppin, <cite class="italique">Mathématiques <a href="/wiki/MPSI" class="mw-redirect" title="MPSI">MPSI</a>: Exercices</cite>, <a href="/wiki/%C3%89ditions_Br%C3%A9al" title="Éditions Bréal">Bréal</a>, <time>2003</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="//books.google.com/books?id=iePuAqUjGoQC&pg=PA33">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 33<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Math%C3%A9matiques+MPSI%3A+Exercices&rft.pub=Br%C3%A9al&rft.aulast=Guinin&rft.aufirst=D.&rft.au=B.+Joppin&rft.date=2003&rft.pages=33&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogarithme"></span></span></span>.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink noprint"><a href="#cite_ref-16">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Ferrier2006"><span class="ouvrage" id="O._Ferrier2006">O. Ferrier, <cite class="italique">Maths pour économistes : L'Analyse en économie</cite>, <abbr class="abbr" title="volume">vol.</abbr> 1, <a href="/wiki/De_Boeck_Sup%C3%A9rieur" class="mw-redirect" title="De Boeck Supérieur">De Boeck Université</a>, <time>2006</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-8041-4354-1" title="Spécial:Ouvrages de référence/978-2-8041-4354-1"><span class="nowrap">978-2-8041-4354-1</span></a>)</small>, <abbr class="abbr" title="page">p.</abbr> 275<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Maths+pour+%C3%A9conomistes&rft.pub=De+Boeck+Universit%C3%A9&rft.stitle=L%27Analyse+en+%C3%A9conomie&rft.aulast=Ferrier&rft.aufirst=O.&rft.date=2006&rft.volume=1&rft.pages=275&rft.isbn=978-2-8041-4354-1&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogarithme"></span></span></span>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink noprint"><a href="#cite_ref-17">↑</a> </span><span class="reference-text">Ne pas confondre avec divers <a href="/wiki/Liste_des_sujets_nomm%C3%A9s_d%27apr%C3%A8s_Leonhard_Euler#Nombres" class="mw-redirect" title="Liste des sujets nommés d'après Leonhard Euler">autres « nombres d'Euler »</a>.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink noprint"><a href="#cite_ref-18">↑</a> </span><span class="reference-text"><a rel="nofollow" class="external text" href="http://www.iso.org/iso/fr/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=31887">ISO 80000-2:2009</a>. <a href="/wiki/Organisation_internationale_de_normalisation" title="Organisation internationale de normalisation">Organisation internationale de normalisation</a>. Consulté le 19 janvier 2012.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink noprint"><a href="#cite_ref-19">↑</a> </span><span class="reference-text"><span class="ouvrage" id="internationale_de_normalisation"><span class="ouvrage" id="Organisation_internationale_de_normalisation"><a href="/wiki/Organisation_internationale_de_normalisation" title="Organisation internationale de normalisation">Organisation internationale de normalisation</a>, « <a rel="nofollow" class="external text" href="https://www.iso.org/fr/standard/64973.html"><cite style="font-style:normal;">ISO 80000-2:2019</cite></a> » <small style="line-height:1em;">(consulté le <time class="nowrap" datetime="2012-09-16" data-sort-value="2012-09-16">16 septembre 2012</time>)</small></span></span>.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink noprint"><a href="#cite_ref-20">↑</a> </span><span class="reference-text"><span class="ouvrage" id="BouvierGeorgeLe_Lionnais2001"><span class="ouvrage" id="Alain_BouvierMichel_GeorgeFrançois_Le_Lionnais2001"><a href="/wiki/Alain_Bouvier" title="Alain Bouvier">Alain <span class="nom_auteur">Bouvier</span></a>, Michel <span class="nom_auteur">George</span> et <a href="/wiki/Fran%C3%A7ois_Le_Lionnais" title="François Le Lionnais">François <span class="nom_auteur">Le Lionnais</span></a>, <cite class="italique">Dictionnaire des mathématiques</cite>, <a href="/wiki/Presses_universitaires_de_France" title="Presses universitaires de France">Presses universitaires de France</a>, <time>2001</time> (<abbr class="abbr" title="première">1<sup>re</sup></abbr> <abbr class="abbr" title="édition">éd.</abbr> 1979), <abbr class="abbr" title="page">p.</abbr> 159<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionnaire+des+math%C3%A9matiques&rft.pub=Presses+universitaires+de+France&rft.aulast=Bouvier&rft.aufirst=Alain&rft.au=George%2C+Michel&rft.au=Le+Lionnais%2C+Fran%C3%A7ois&rft.date=2001&rft.pages=159&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogarithme"></span></span></span>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=16" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=16" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint boite-grise boite-a-droite" style="text-align:left;"> <div style="float:left;"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Circle-icons-frames.svg/45px-Circle-icons-frames.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Circle-icons-frames.svg/68px-Circle-icons-frames.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Circle-icons-frames.svg/90px-Circle-icons-frames.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div style="margin-left:60px;">Une <a href="/wiki/Aide:Cat%C3%A9gorie" title="Aide:Catégorie">catégorie</a> est consacrée à ce sujet : <i><a href="/wiki/Cat%C3%A9gorie:Logarithme" title="Catégorie:Logarithme">Logarithme</a></i>.</div> <div style="clear:left;"></div> </div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> <p class="titre">Sur les autres projets Wikimedia :</p> <ul class="noarchive plainlinks"> <li class="commons"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Logarithm?uselang=fr">Les logarithmes</a>, sur <span class="project">Wikimedia Commons</span></li><li class="wiktionary"><a href="https://fr.wiktionary.org/wiki/logarithme" class="extiw" title="wikt:logarithme">logarithme</a>, <span class="nowrap">sur le <span class="project">Wiktionnaire</span></span></li><li class="wikiversity"><a href="https://fr.wikiversity.org/wiki/Fonction_logarithme" class="extiw" title="v:Fonction logarithme">Logarithme</a>, <span class="nowrap">sur <span class="project">Wikiversity</span></span></li><li class="wikibooks"><a href="https://fr.wikibooks.org/wiki/Photographie/Math%C3%A9matiques" class="extiw" title="b:Photographie/Mathématiques">Photographie/Mathématiques (sections "Découverte des logarithmes" et "Que fait-on avec les logarithmes ?")</a>, <span class="nowrap">sur <span class="project">Wikibooks</span></span></li> </ul> </div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=17" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=17" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="column-width:15em;column-gap:1em;" class="colonnes"> <ul><li><a href="/wiki/Logarithme_complexe" title="Logarithme complexe">Logarithme complexe</a></li> <li><a href="/wiki/Fonction_polylogarithme" title="Fonction polylogarithme">Fonction polylogarithme</a></li> <li><a href="/wiki/Fonction_holomorphe" title="Fonction holomorphe">Fonction holomorphe</a></li> <li><a href="/wiki/Loi_de_Benford" title="Loi de Benford">Loi de Benford</a></li></ul> </div> <div class="mw-heading mw-heading4"><h4 id="Applications_pratiques">Applications pratiques</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=18" title="Modifier la section : Applications pratiques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=18" title="Modifier le code source de la section : Applications pratiques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="column-width:15em;column-gap:1em;" class="colonnes"> <ul><li><a href="/wiki/R%C3%A8gle_%C3%A0_calcul" title="Règle à calcul">Règle à calcul</a></li> <li><a href="/wiki/%C3%89chelle_logarithmique" title="Échelle logarithmique">Échelle logarithmique</a></li> <li><a href="/wiki/Table_de_logarithmes" title="Table de logarithmes">Table de logarithmes</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Liens_externes">Liens externes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme&veaction=edit&section=19" title="Modifier la section : Liens externes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme&action=edit&section=19" title="Modifier le code source de la section : Liens externes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p class="mw-empty-elt"> </p> <ul><li class="mw-empty-elt"></li> <li class="mw-empty-elt"></li> <li><div class="liste-horizontale"><span class="wd_identifiers">Notices dans des dictionnaires ou encyclopédies généralistes<span class="noprint wikidata-linkback skin-invert"><span class="mw-valign-baseline noviewer" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q11197?uselang=fr#identifiers" title="Voir et modifier les données sur Wikidata"><img alt="Voir et modifier les données sur Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></span> : <ul><li><a rel="nofollow" class="external text" href="https://www.britannica.com/science/logarithm"><i>Britannica</i></a></li> <li><a rel="nofollow" class="external text" href="https://denstoredanske.lex.dk//logaritme/"><i>Den Store Danske Encyklopædi</i></a></li> <li><a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/logaritmo_(Enciclopedia-Italiana)/"><i>Enciclopedia italiana</i></a></li> <li><a rel="nofollow" class="external text" href="http://www.sapere.it/enciclopedia/logaritmo.html"><i>Enciclopedia De Agostini</i></a></li> <li><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3933548"><i>Internetowa encyklopedia PWN</i></a></li> <li><a rel="nofollow" class="external text" href="https://www.larousse.fr/encyclopedie/images/Logarithmes/1012284"><i>Larousse</i></a></li> <li><a rel="nofollow" class="external text" href="https://snl.no/logaritme"><i>Store norske leksikon</i></a></li> <li><a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/logaritmo"><i>Treccani</i></a></li> </ul></div></li> <li><div class="liste-horizontale"><span class="wd_identifiers"><a href="/wiki/Autorit%C3%A9_(sciences_de_l%27information)" title="Autorité (sciences de l'information)">Notices d'autorité</a><span class="noprint wikidata-linkback skin-invert"><span class="mw-valign-baseline noviewer" typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q11197?uselang=fr#identifiers" title="Voir et modifier les données sur Wikidata"><img alt="Voir et modifier les données sur Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></span></span> : <ul><li><span class="nowrap uid noarchive"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11941516p">BnF</a></span> (<span class="nowrap uid noarchive"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11941516p">données</a></span>)</li> <li><span class="nowrap uid noarchive"><a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/sh85078091">LCCN</a></span></li> <li><span class="nowrap uid noarchive"><a rel="nofollow" class="external text" href="http://d-nb.info/gnd/4168047-9">GND</a></span></li> <li><span class="nowrap uid noarchive"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00572566">Japon</a></span></li> <li><span class="nowrap uid noarchive"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX527539">Espagne</a></span></li> <li><span class="nowrap uid noarchive"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007533701405171">Israël</a></span></li> </ul></div></li> <li>Simone Trompler, <a rel="nofollow" class="external text" href="https://sonocreatica.org.ve/wp-content/uploads/2018/03/Histoire_Logarithme.pdf">Histoire des logarithmes</a>, publié en ligne en 2002 par l’<a href="/wiki/Universit%C3%A9_libre_de_Bruxelles" title="Université libre de Bruxelles">Université libre de Bruxelles</a></li></ul> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Fonctions_math%C3%A9matiques_usuelles" title="Modèle:Palette Fonctions mathématiques usuelles"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Fonctions_math%C3%A9matiques_usuelles&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Fonction_(math%C3%A9matiques)" title="Fonction (mathématiques)">Fonctions mathématiques usuelles</a></div></th> </tr> <tr> <th class="navbox-group" style="width:200px"><a href="/wiki/Fonction_alg%C3%A9brique" title="Fonction algébrique">Fonction algébrique</a> <a href="/wiki/Fonction_rationnelle" title="Fonction rationnelle">rationnelle</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Fonction_polynomiale" title="Fonction polynomiale">Fonction polynomiale</a></li> <li><a href="/wiki/Fraction_rationnelle" title="Fraction rationnelle">Fonction fractionnaire</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:200px">Fonction algébrique irrationnelle</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Fonction_puissance" title="Fonction puissance">Fonction puissance</a> / <a href="/wiki/Racine_d%27un_nombre" title="Racine d'un nombre">Fonction racine</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:200px"><a href="/wiki/Fonction_transcendante" title="Fonction transcendante">Fonction transcendante</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a class="mw-selflink selflink">Fonction logarithmique</a> / <a href="/wiki/Exponentielle_de_base_a" title="Exponentielle de base a">Fonction exponentielle de base a</a> <ul><li><a href="/wiki/Logarithme_n%C3%A9p%C3%A9rien" title="Logarithme népérien">Fonction logarithme naturel</a> / <a href="/wiki/Fonction_exponentielle" title="Fonction exponentielle">Fonction exponentielle</a></li></ul></li> <li><a href="/wiki/Fonction_trigonom%C3%A9trique" title="Fonction trigonométrique">Fonction circulaire</a> / <a href="/wiki/Fonction_circulaire_r%C3%A9ciproque" title="Fonction circulaire réciproque">Fonction circulaire réciproque</a></li> <li><a href="/wiki/Fonction_hyperbolique" title="Fonction hyperbolique">Fonction hyperbolique</a> / <a href="/wiki/Fonction_hyperbolique#Applications_réciproques" title="Fonction hyperbolique">Fonction hyperbolique réciproque</a></li> <li><a href="/wiki/Fonction_elliptique" title="Fonction elliptique">Fonction elliptique</a> / <a href="/wiki/Int%C3%A9grale_elliptique" title="Intégrale elliptique">Fonction intégrale elliptique</a></li></ul> </div></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Analyse" title="Portail de l'analyse"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/24px-Nuvola_apps_kmplot.svg.png" decoding="async" width="24" height="24" class="mw-file-element" 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