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class="vector-toc-link" href="#Mantisse_et_caractéristique"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Mantisse et caractéristique</span> </div> </a> <ul id="toc-Mantisse_et_caractéristique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Usage" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Usage"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Usage</span> </div> </a> <button aria-controls="toc-Usage-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 Usage</span> </button> <ul id="toc-Usage-sublist" class="vector-toc-list"> <li id="toc-Calculer_avec_une_table_de_logarithmes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calculer_avec_une_table_de_logarithmes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Calculer avec une table de logarithmes</span> </div> </a> <ul id="toc-Calculer_avec_une_table_de_logarithmes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_règle_à_calcul" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_règle_à_calcul"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>La règle à calcul</span> </div> </a> <ul id="toc-La_règle_à_calcul-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Les_échelles_logarithmiques" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Les_échelles_logarithmiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Les échelles logarithmiques</span> </div> </a> <ul id="toc-Les_échelles_logarithmiques-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Le_pH" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Le_pH"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Le pH</span> </div> </a> <ul id="toc-Le_pH-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Les_décibels" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Les_décibels"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Les décibels</span> </div> </a> <ul id="toc-Les_décibels-sublist" class="vector-toc-list"> </ul> </li> </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">4</span> <span>Notes et références</span> </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="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 décimal</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 35 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-35" 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">35 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D9%88%D8%BA%D8%A7%D8%B1%D9%8A%D8%AA%D9%85_%D8%B9%D8%B4%D8%B1%D9%8A" title="لوغاريتم عشري – arabe" lang="ar" hreflang="ar" data-title="لوغاريتم عشري" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Onluq_loqarifm" title="Onluq loqarifm – azerbaïdjanais" lang="az" hreflang="az" data-title="Onluq loqarifm" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D1%81%D0%B5%D1%82%D0%B8%D1%87%D0%B5%D0%BD_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D1%8A%D0%BC" title="Десетичен логаритъм – bulgare" lang="bg" hreflang="bg" data-title="Десетичен логаритъм" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Logaritme_decimal" title="Logaritme decimal – catalan" lang="ca" hreflang="ca" data-title="Logaritme decimal" 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%DB%8C_%D8%AF%DB%95%DB%8C%DB%8C" title="لۆگاریتمی دەیی – sorani" lang="ckb" hreflang="ckb" data-title="لۆگاریتمی دەیی" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D1%83%D0%BD%D0%BD%C4%83%D0%BB%D0%BB%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Вуннăлла логарифм – tchouvache" lang="cv" hreflang="cv" data-title="Вуннăлла логарифм" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dekadischer_Logarithmus" title="Dekadischer Logarithmus – allemand" lang="de" hreflang="de" data-title="Dekadischer Logarithmus" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Common_logarithm" title="Common logarithm – anglais" lang="en" hreflang="en" data-title="Common logarithm" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Logaritmo_decimal" title="Logaritmo decimal – espagnol" lang="es" hreflang="es" data-title="Logaritmo decimal" 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/K%C3%BCmnendlogaritm" title="Kümnendlogaritm – estonien" lang="et" hreflang="et" data-title="Kümnendlogaritm" 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_hamartar" title="Logaritmo hamartar – basque" lang="eu" hreflang="eu" data-title="Logaritmo hamartar" data-language-autonym="Euskara" data-language-local-name="basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%84%DA%AF%D8%A7%D8%B1%DB%8C%D8%AA%D9%85_%D8%B1%D8%A7%DB%8C%D8%AC" title="لگاریتم رایج – persan" lang="fa" hreflang="fa" data-title="لگاریتم رایج" 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/Briggsin_logaritmi" title="Briggsin logaritmi – finnois" lang="fi" hreflang="fi" data-title="Briggsin logaritmi" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Tjiiner_logarithmus" title="Tjiiner logarithmus – frison septentrional" lang="frr" hreflang="frr" data-title="Tjiiner logarithmus" data-language-autonym="Nordfriisk" data-language-local-name="frison septentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A4%A3_%E0%A4%B2%E0%A4%98%E0%A5%81%E0%A4%97%E0%A4%A3%E0%A4%95" title="साधारण लघुगणक – hindi" lang="hi" hreflang="hi" data-title="साधारण लघुगणक" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A1%D5%BD%D5%B6%D5%B8%D6%80%D5%A4%D5%A1%D5%AF%D5%A1%D5%B6_%D5%AC%D5%B8%D5%A3%D5%A1%D6%80%D5%AB%D5%A9%D5%B4%D5%B6%D5%A5%D6%80" title="Տասնորդական լոգարիթմներ – arménien" lang="hy" hreflang="hy" data-title="Տասնորդական լոգարիթմներ" data-language-autonym="Հայերեն" data-language-local-name="arménien" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logaritma_umum" title="Logaritma umum – indonésien" lang="id" hreflang="id" data-title="Logaritma umum" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B8%B8%E7%94%A8%E5%AF%BE%E6%95%B0" title="常用対数 – japonais" lang="ja" hreflang="ja" data-title="常用対数" 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-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9E%D0%BD%D0%B4%D1%8B%D2%9B_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Ондық логарифм – kazakh" lang="kk" hreflang="kk" data-title="Ондық 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/%EC%83%81%EC%9A%A9%EB%A1%9C%EA%B7%B8" title="상용로그 – coréen" lang="ko" hreflang="ko" data-title="상용로그" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logaritma_biasa" title="Logaritma biasa – malais" lang="ms" hreflang="ms" data-title="Logaritma biasa" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Briggse_logaritme" title="Briggse logaritme – néerlandais" lang="nl" hreflang="nl" data-title="Briggse logaritme" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Logaritme_decimau" title="Logaritme decimau – occitan" lang="oc" hreflang="oc" data-title="Logaritme decimau" data-language-autonym="Occitan" data-language-local-name="occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logarytm_dziesi%C4%99tny" title="Logarytm dziesiętny – polonais" lang="pl" hreflang="pl" data-title="Logarytm dziesiętny" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Logaritmo_comum" title="Logaritmo comum – portugais" lang="pt" hreflang="pt" data-title="Logaritmo comum" 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_zecimal" title="Logaritm zecimal – roumain" lang="ro" hreflang="ro" data-title="Logaritm zecimal" data-language-autonym="Română" data-language-local-name="roumain" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D1%81%D1%8F%D1%82%D0%B8%D1%87%D0%BD%D1%8B%D0%B9_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Десятичный логарифм – russe" lang="ru" hreflang="ru" data-title="Десятичный логарифм" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%83%E0%B7%8F%E0%B6%B8%E0%B7%8F%E0%B6%B1%E0%B7%8A%E2%80%8D%E0%B6%BA_%E0%B6%BD%E0%B6%9D%E0%B7%94%E0%B6%9C%E0%B6%AB%E0%B6%9A%E0%B6%BA" title="සාමාන්‍ය ලඝුගණකය – cingalais" lang="si" hreflang="si" data-title="සාමාන්‍ය ලඝුගණකය" data-language-autonym="සිංහල" data-language-local-name="cingalais" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AF%8A%E0%AE%A4%E0%AF%81_%E0%AE%AE%E0%AE%9F%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%88" title="பொது மடக்கை – tamoul" lang="ta" hreflang="ta" data-title="பொது மடக்கை" data-language-autonym="தமிழ்" data-language-local-name="tamoul" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Adi_logaritma" title="Adi logaritma – turc" lang="tr" hreflang="tr" data-title="Adi logaritma" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D1%81%D1%8F%D1%82%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Десятковий логарифм – ukrainien" lang="uk" hreflang="uk" data-title="Десятковий логарифм" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Brigs_logarifmlari" title="Brigs logarifmlari – ouzbek" lang="uz" hreflang="uz" data-title="Brigs logarifmlari" 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 mw-list-item"><a href="https://vi.wikipedia.org/wiki/Logarit_th%C3%B4ng_th%C6%B0%E1%BB%9Dng" title="Logarit thông thường – vietnamien" lang="vi" hreflang="vi" data-title="Logarit thông thường" 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-zh 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src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Logb10.svg/440px-Logb10.svg.png" decoding="async" width="440" height="326" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Logb10.svg/660px-Logb10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Logb10.svg/880px-Logb10.svg.png 2x" data-file-width="2510" data-file-height="1859" /></a><figcaption>Représentation graphique du logarithme décimal dans un repère orthogonal</figcaption></figure> <p>Le <b>logarithme décimal</b> ou <b>log<sub>10</sub></b> ou simplement <b>log</b> (parfois appelé <b>logarithme vulgaire</b>) est le <a href="/wiki/Logarithme" title="Logarithme">logarithme</a> de base <a href="/wiki/10_(nombre)" title="10 (nombre)">dix</a>. Il est défini pour tout réel strictement positif <i>x</i>. </p><p>Le logarithme décimal est la fonction <a href="/wiki/Continuit%C3%A9_(math%C3%A9matiques)" title="Continuité (mathématiques)">continue</a> qui transforme un produit en somme et qui vaut 1 en 10. </p><p>Le logarithme décimal est la <a href="/wiki/Bijection_r%C3%A9ciproque" title="Bijection réciproque">fonction réciproque</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 f(x)=10^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=10^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aba6d31a7420b6c2316c2cad76675b9a22c06d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.013ex; height:2.843ex;" alt="{\displaystyle f(x)=10^{x}}"></span> : <span style="display: block; margin-left:1.6em;">pour <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span>, 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 y=\log _{10}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\log _{10}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc905503c28162278e6adba2ab70468f9551ad19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.241ex; height:2.843ex;" alt="{\displaystyle y=\log _{10}(x)}"></span> alors <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=10^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=10^{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc54a617a4377ac2bd96a6ed69a33dd1a125ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.802ex; height:2.343ex;" alt="{\displaystyle x=10^{y}}"></span>.</span> </p><p>La norme ISO 80000-2<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> indique que log<sub>10</sub> devrait être noté <b>lg</b>, mais cette notation est rarement utilisée. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Histoire">Histoire</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=1" title="Modifier la section : Histoire" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&action=edit&section=1" title="Modifier le code source de la section : Histoire"><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">Articles connexes : <a href="/wiki/Table_de_logarithmes" title="Table de logarithmes">Table de logarithmes</a> et <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> <p>Les logarithmes décimaux sont parfois appelés <b>logarithmes de Briggs</b>. <a href="/wiki/Henry_Briggs" title="Henry Briggs">Henry Briggs</a>, <a href="/wiki/Math%C3%A9maticien" title="Mathématicien">mathématicien</a> britannique du <abbr class="abbr" title="17ᵉ siècle"><span class="romain">XVII</span><sup style="font-size:72%">e</sup></abbr> siècle, est l'auteur de <a href="/wiki/Table_de_logarithmes" title="Table de logarithmes">tables de logarithmes</a> décimaux publiées à <a href="/wiki/Londres" title="Londres">Londres</a> en <a href="/wiki/1624" title="1624">1624</a>, dans un traité intitulé <i>Arithmetica Logarithmetica</i>. </p><p>Avant <a href="/wiki/1970" title="1970">1970</a>, les <a href="/wiki/Calculatrice" title="Calculatrice">calculatrices</a> électroniques n'étaient pas encore d'un usage très répandu, et elles étaient assez volumineuses. Pour effectuer des produits ou des quotients, on utilisait encore des tables de logarithmes de base dix ou des <a href="/wiki/R%C3%A8gle_%C3%A0_calcul" title="Règle à calcul">règles à calcul</a>, et les calculs étaient effectués « à la main » sur papier. </p><p>Les logarithmes de base dix ou logarithmes décimaux étaient appelés <b>logarithmes vulgaires</b>, par opposition aux logarithmes de base <a href="/wiki/E_(nombre)" title="E (nombre)">e</a>, dits <a href="/wiki/Logarithme_naturel" class="mw-redirect" title="Logarithme naturel">logarithmes naturels</a>, népériens ou hyperboliques. </p><p>Dans <i>An Introduction to the Theory of Numbers</i>, <a href="/wiki/Godfrey_Harold_Hardy" title="Godfrey Harold Hardy">Godfrey Harold Hardy</a> écrit une note : </p> <style data-mw-deduplicate="TemplateStyles:r201232302">.mw-parser-output .container{display:flex;flex-wrap:wrap;margin:1em 40px;overflow:hidden;justify-content:space-between}@media(max-width:768px){.mw-parser-output .item{flex-grow:1;flex-shrink:1;flex-basis:auto}}@media(min-width:769px){.mw-parser-output .item{flex-basis:45%}}.mw-parser-output .blockquote{margin:0}.mw-parser-output .author{margin:-0.5em 0 1em 60px}</style><div class="container"> <div class="item"> <blockquote class="blockquote" lang="en"><p>« <i>log <i>x</i> is, of course, the 'Napierian' logarithm of <i>x</i>, to base <i>e</i>. 'Common' logarithms have no mathematical interest<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>.</i> »</p></blockquote> </div> <div class="item"> <blockquote class="blockquote"><p>« log <i>x</i> est, bien sûr, le logarithme « néperien » de <i>x</i>, de base <i>e</i>. Les logarithmes « vulgaires » n'ont pas d'intérêt mathématique. »</p></blockquote> </div> </div> <div class="mw-heading mw-heading2"><h2 id="Mantisse_et_caractéristique"><span id="Mantisse_et_caract.C3.A9ristique"></span>Mantisse et caractéristique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=2" title="Modifier la section : Mantisse et caractéristique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&action=edit&section=2" title="Modifier le code source de la section : Mantisse et caractéristique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les logarithmes des puissances entières de 10 se calculent aisément en utilisant la règle de conversion d'un produit en somme : <span style="display: block; margin-left:1.6em;"><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)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(10)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa5afb256560636c0f067ba86879e9b11188b4b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.367ex; height:2.843ex;" alt="{\displaystyle \log(10)=1}"></span><br /> <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(100)=\log(10\times 10)=\log(10)+\log(10)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>100</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo>×</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(100)=\log(10\times 10)=\log(10)+\log(10)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d03cadc127ae49d5e70fa7ec586bec5fc74ad4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.05ex; height:2.843ex;" alt="{\displaystyle \log(100)=\log(10\times 10)=\log(10)+\log(10)=2}"></span><br /> <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(1\,000)=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mn>000</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(1\,000)=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62d543df1d97a874361b61326870e79dd2c483d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.079ex; height:2.843ex;" alt="{\displaystyle \log(1\,000)=3}"></span><br /> <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^{n})=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <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(10^{n})=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a9d0a569b5fb74d6e9087eefc0e2bc1524a1dc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.818ex; height:2.843ex;" alt="{\displaystyle \log(10^{n})=n}"></span><br /> <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(0{,}1)=\log \left({\frac {1}{10}}\right)=-\log(10)=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(0{,}1)=\log \left({\frac {1}{10}}\right)=-\log(10)=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9761456dde1c814754bd3c1b7b745d5669345b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.874ex; height:6.176ex;" alt="{\displaystyle \log(0{,}1)=\log \left({\frac {1}{10}}\right)=-\log(10)=-1}"></span><br /> <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(0{,}01)=-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>01</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(0{,}01)=-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1118f5d0ba2920e8c31f9d011bf36d11dd5752ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.984ex; height:2.843ex;" alt="{\displaystyle \log(0{,}01)=-2}"></span><br /> <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(0{,}001)=-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0,001</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(0{,}001)=-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f9bb3aabbbfeaa05dd2e95f474205923c65169d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.147ex; height:2.843ex;" alt="{\displaystyle \log(0{,}001)=-3}"></span>.</span> Les propriétés arithmétiques des logarithmes permettent de déduire la valeur de tout logarithme pourvu que soient connus les logarithmes de tous les nombres compris entre 1 et 10 (exclu). En effet, tout nombre <i>x</i> peut s'écrire sous la forme <i>a</i> × 10<sup><i>n</i></sup> où <i>a</i> est un nombre compris entre 1 et 10 (exclu). Cette écriture s'appelle la <a href="/wiki/Notation_scientifique" title="Notation scientifique">notation scientifique</a> de <i>x</i> × 10<sup><i>n</i></sup> représente alors l'ordre de grandeur du nombre <i>x</i>. Par exemple <span style="display: block; margin-left:1.6em;"><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 120=1{,}2\times 10^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>120</mn> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>2</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 120=1{,}2\times 10^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bf55405b041b622dcb2cb137fcf69e6e216d51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.777ex; height:3.009ex;" alt="{\displaystyle 120=1{,}2\times 10^{2}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}00314=3{,}14\times 10^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0,003</mn> <mn>14</mn> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>14</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0{,}00314=3{,}14\times 10^{-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3267548628d7d7f08ad168e24decb2c40922ea04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.352ex; height:3.009ex;" alt="{\displaystyle 0{,}00314=3{,}14\times 10^{-3}}"></span>.</span> Le passage au logarithme décimal va alors mettre en évidence les deux éléments de l'écriture scientifique du nombre <span style="display: block; margin-left:1.6em;"><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(120)=\log(1{,}2)+\log(10^{2})=\log(1{,}2)+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>120</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(120)=\log(1{,}2)+\log(10^{2})=\log(1{,}2)+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef3f09658a132784ce7d803aed0e96aa6a0c4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.974ex; height:3.176ex;" alt="{\displaystyle \log(120)=\log(1{,}2)+\log(10^{2})=\log(1{,}2)+2}"></span><br /> <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(0{,}00314)=\log(3{,}14)+\log(10^{-3})=\log(3{,}14)-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0,003</mn> <mn>14</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>14</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>14</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(0{,}00314)=\log(3{,}14)+\log(10^{-3})=\log(3{,}14)-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c0c11547935a69bdb4472d97d8b11fe20d955a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.712ex; height:3.176ex;" alt="{\displaystyle \log(0{,}00314)=\log(3{,}14)+\log(10^{-3})=\log(3{,}14)-3}"></span><br /> <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(x)=\log(a\times 10^{n})=n+\log(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(x)=\log(a\times 10^{n})=n+\log(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7f1fb0a8456831b76a6536e1cb6b3069d1d040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.948ex; height:2.843ex;" alt="{\displaystyle \log(x)=\log(a\times 10^{n})=n+\log(a)}"></span>.</span> Puisque la fonction log est croissante, pour tout réel <i>a</i> compris entre 1 et 10 (exclu), log(<i>a</i>) est compris entre 0 et 1. L'entier relatif <i>n</i> est donc la <a href="/wiki/Partie_enti%C3%A8re" class="mw-redirect" title="Partie entière">partie entière</a> de log(<i>x</i>) et log(<i>a</i>) la <a href="/wiki/Partie_d%C3%A9cimale" class="mw-redirect" title="Partie décimale">partie décimale</a> à ajouter à <i>n</i> pour obtenir log(<i>x</i>). </p><p>La partie entière de log(<i>x</i>) est appelée <b>caractéristique</b> du log. </p><p>La partie décimale à rajouter à la partie entière s'appelle <b>mantisse</b>. </p><p>On fera attention à l'écriture du logarithme des nombres plus petits que 1 : <span style="display: block; margin-left:1.6em;"><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(0{,}00314)=-3+\log(3{,}14)\approx -3+0{,}497}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0,003</mn> <mn>14</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>14</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>0,497</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(0{,}00314)=-3+\log(3{,}14)\approx -3+0{,}497}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9b63d659732c060f147395c44391369a9c1261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.433ex; height:2.843ex;" alt="{\displaystyle \log(0{,}00314)=-3+\log(3{,}14)\approx -3+0{,}497}"></span><br /> <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(0{,}00314)\approx -2{,}503.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0,003</mn> <mn>14</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mo>−</mo> <mn>2,503.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(0{,}00314)\approx -2{,}503.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb18fd48f2091601841c6672df31d5832446674" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.253ex; height:2.843ex;" alt="{\displaystyle \log(0{,}00314)\approx -2{,}503.}"></span></span> La deuxième écriture, qui semble plus naturelle, ne permet pas de retrouver rapidement la caractéristique (−3) et la mantisse (0,497). On préfère alors utiliser la première écriture que l'on note souvent <span style="display: block; margin-left:1.6em;"><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(0{,}00314)\approx {\overline {3}}{,}497}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>0,003</mn> <mn>14</mn> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>497</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(0{,}00314)\approx {\overline {3}}{,}497}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257671e09543c86d74be96ef3e66607b023f4b1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.913ex; height:3.343ex;" alt="{\displaystyle \log(0{,}00314)\approx {\overline {3}}{,}497}"></span>.</span> La lecture du logarithme d'un nombre permet alors aisément de déterminer son ordre de grandeur : <span style="display: block; margin-left:1.6em;">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 \log(x)=5{,}3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(x)=5{,}3.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0f7f189deac91163cdbe813c6bc7a76fa7824b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.828ex; height:2.843ex;" alt="{\displaystyle \log(x)=5{,}3.}"></span></span> Sa caractéristique est 5 donc <i>x</i> est de la forme <i>a</i> × 10<sup>5</sup>. Sa mantisse est 0,3 qui est proche de log(2). <i>x</i> est donc proche de 2 × 10<sup>5</sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Usage">Usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=3" title="Modifier la section : Usage" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&action=edit&section=3" title="Modifier le code source de la section : Usage"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le développement des calculatrices de poche a fait perdre aux logarithmes leur principal intérêt de simplification des calculs. Ils restent cependant très présents en physique quand il s'agit d'appréhender des quantités pouvant varier de 10<sup>−10</sup> à 10<sup>10</sup>. C'est ainsi qu'on les retrouve dans le calcul des pH (<a href="/wiki/Potentiel_hydrog%C3%A8ne" title="Potentiel hydrogène">potentiel hydrogène</a>), des <a href="/wiki/D%C3%A9cibel" title="Décibel">décibels</a>, … </p> <div class="mw-heading mw-heading3"><h3 id="Calculer_avec_une_table_de_logarithmes">Calculer avec une table de logarithmes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=4" title="Modifier la section : Calculer avec une table de logarithmes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&action=edit&section=4" title="Modifier le code source de la section : Calculer avec une table de logarithmes"><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/Table_de_logarithmes" title="Table de logarithmes">table de logarithmes</a>.</div></div> <p>L'idée directrice est de remplacer, pour l'utilisateur, les multiplications par des additions, les divisions par des soustractions, les puissances par des produits, les racines nièmes par des divisions par <i>n</i>. </p> <div style="margin:0.5em 2em;"><strong>Exemple 1 :</strong> <div style="padding-left:2em; border-left:1px dotted #999;"> <p>En supposant que <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=435{,}728}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>435,728</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=435{,}728}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0a771391c6a8e08fb6019cb65dabc0628c4f67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.05ex; height:2.509ex;" alt="{\displaystyle x=435{,}728}"></span> et <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 y=1{,}6275}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>1,627</mn> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=1{,}6275}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0444e8899622cae908c5cd3de3fb2e5ec79803c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.713ex; height:2.509ex;" alt="{\displaystyle y=1{,}6275}"></span> comment effectuer, sans calculatrice, le 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 x\;y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thickmathspace" /> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\;y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccbfccf99cce4b582be33bed6c4ee9341fa1a129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.13ex; height:2.009ex;" alt="{\displaystyle x\;y}"></span> ? <span style="display: block; margin-left:1.6em;">On calcule <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(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4157d3b51ac7b147fca145d431d58ec92abc1f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.111ex; height:2.843ex;" alt="{\displaystyle \log(x)}"></span><br /> <span style="display: block; margin-left:1.6em;"><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=4{,}35728\times 10^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>4,357</mn> <mn>28</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=4{,}35728\times 10^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40b98812bec25100d0922b6a4f2299105b6d4a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.269ex; height:3.009ex;" alt="{\displaystyle x=4{,}35728\times 10^{2}}"></span> donc la caractéristique est 2, la mantisse se lit dans une table de logarithme : 0,6392<br /> <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(x)=2{,}6392}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2,639</mn> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(x)=2{,}6392}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21ceb09ff5c927de36feec2ec711f2c3da232817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.668ex; height:2.843ex;" alt="{\displaystyle \log(x)=2{,}6392}"></span></span> On calcule log(<i>y</i>), caractéristique 0, mantisse 0,2115<br /> <span style="display: block; margin-left:1.6em;"><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(y)=0,2115}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>2115</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(y)=0,2115}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78a2eb34864bc5e1f08bdb5c371b9790de79d0c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.881ex; height:2.843ex;" alt="{\displaystyle \log(y)=0,2115}"></span>.</span></span> Il suffit de calculer <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(x\;y)=\log(x)+\log(y)=2{,}8507}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mspace width="thickmathspace" /> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2,850</mn> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(x\;y)=\log(x)+\log(y)=2{,}8507}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b456b98a7187ad30fc9ddb64fd0f9e3cbb700fe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.455ex; height:2.843ex;" alt="{\displaystyle \log(x\;y)=\log(x)+\log(y)=2{,}8507}"></span>, d'isoler la caractéristique 2 et la mantisse 0,8507 qui par lecture inverse dans la table de log donne 7,091. </p><p>Le 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 x\;y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mspace width="thickmathspace" /> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\;y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccbfccf99cce4b582be33bed6c4ee9341fa1a129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.13ex; height:2.009ex;" alt="{\displaystyle x\;y}"></span> est donc environ <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 7{,}091\times 10^{2}=709{,}1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7,091</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>709</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7{,}091\times 10^{2}=709{,}1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72bb5ba3cf06a22a7e976a2ff2e8197d20e5d2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.911ex; height:3.009ex;" alt="{\displaystyle 7{,}091\times 10^{2}=709{,}1}"></span>. </p> </div></div> <div style="margin:0.5em 2em;"><strong>Exemple 2 :</strong> <div style="padding-left:2em; border-left:1px dotted #999;"> <p>En prenant toujours ces deux nombres, on peut tout aussi facilement calculer une valeur approchée de la racine cubique de leur quotient <span style="display: block; margin-left:1.6em;"><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 \left({\sqrt[{3}]{\frac {x}{y}}}\right)={\frac {1}{3}}{\Big (}\log(x)-\log(y){\Big )}={\frac {2{,}6392-0{,}2115}{3}}={\frac {2{,}4277}{3}}=0{,}8092}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2,639</mn> <mn>2</mn> <mo>−</mo> <mn>0,211</mn> <mn>5</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2,427</mn> <mn>7</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>=</mo> <mn>0,809</mn> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log \left({\sqrt[{3}]{\frac {x}{y}}}\right)={\frac {1}{3}}{\Big (}\log(x)-\log(y){\Big )}={\frac {2{,}6392-0{,}2115}{3}}={\frac {2{,}4277}{3}}=0{,}8092}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf6963378d285c3610fddaa2e778de59c2f1b51e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:73.934ex; height:7.509ex;" alt="{\displaystyle \log \left({\sqrt[{3}]{\frac {x}{y}}}\right)={\frac {1}{3}}{\Big (}\log(x)-\log(y){\Big )}={\frac {2{,}6392-0{,}2115}{3}}={\frac {2{,}4277}{3}}=0{,}8092}"></span>.</span> La caractéristique est donc nulle, la mantisse est 0,8092 qui, par lecture inverse, donne 6,445. </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[{3}]{\frac {x}{y}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{\frac {x}{y}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19074aee265076bc8c1977057d12c0a6069112b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:4.49ex; height:6.176ex;" alt="{\displaystyle {\sqrt[{3}]{\frac {x}{y}}}}"></span> est donc environ égal à 6,445. </p> </div></div> <div class="mw-heading mw-heading3"><h3 id="La_règle_à_calcul"><span id="La_r.C3.A8gle_.C3.A0_calcul"></span>La règle à calcul</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=5" title="Modifier la section : La règle à calcul" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&action=edit&section=5" title="Modifier le code source de la section : La règle à calcul"><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/R%C3%A8gle_%C3%A0_calcul" title="Règle à calcul">règle à calcul</a>.</div></div> <p>Le principe de la règle à calcul est analogue à celui précédemment décrit. La précision sera seulement moindre. </p><p>Sur la règle à calcul sont placés les logarithmes des nombres de 1 à 10. </p><p>Pour effectuer le produit de <i>x y</i> = 436 × 1,63, on effectue, grâce à la règle à calcul, le produit 4,36 × 1,63 en ajoutant les longueurs correspondant à log(4,36) et log(1,63), on obtient environ 7,1. </p><p>Le produit de <i>x y</i> est donc environ 7,1 × 10<sup>2</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Les_échelles_logarithmiques"><span id="Les_.C3.A9chelles_logarithmiques"></span>Les échelles logarithmiques</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=6" title="Modifier la section : Les échelles logarithmiques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&action=edit&section=6" title="Modifier le code source de la section : Les échelles logarithmiques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elles sont utilisées pour représenter des phénomènes pouvant varier par exemple 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 10^{-10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>10</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7db6cb0ee6e78693aadaa8206396bc041b091c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.48ex; height:2.676ex;" alt="{\displaystyle 10^{-10}}"></span> 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 10^{10}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b086010e3cc3b0a4e22c858243d32ec1cc648e6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.201ex; height:2.676ex;" alt="{\displaystyle 10^{10}}"></span>. Elles permettent d'amplifier les variations des valeurs proches de 0 et de rendre moins importantes les variations pour les grands nombres, en mettant en évidence plutôt les variations relatives. </p><p>L'utilisation des échelles logarithmiques est détaillée dans les articles <a href="/wiki/%C3%89chelle_logarithmique" title="Échelle logarithmique">Échelle logarithmique</a>, <a href="/wiki/Rep%C3%A8re_semi-logarithmique" title="Repère semi-logarithmique">Repère semi-logarithmique</a> et <a href="/wiki/Rep%C3%A8re_log-log" title="Repère log-log">Repère log-log</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Le_pH">Le pH</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=7" title="Modifier la section : Le pH" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&action=edit&section=7" title="Modifier le code source de la section : Le pH"><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/Potentiel_hydrog%C3%A8ne" title="Potentiel hydrogène">Potentiel hydrogène</a>.</div></div> <p>Le pH d'une solution donne le <a href="/wiki/Cologarithme" title="Cologarithme">cologarithme</a> de sa concentration en <a href="/wiki/Ion_oxonium" title="Ion oxonium">ions oxonium</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 \mathrm {pH} =-\log {\big [}\mathrm {H} _{3}\mathrm {O} ^{+}{\big ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">H</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {pH} =-\log {\big [}\mathrm {H} _{3}\mathrm {O} ^{+}{\big ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ba879be071a53fcbd4e77de3f5bdd7007303ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.744ex; height:3.176ex;" alt="{\displaystyle \mathrm {pH} =-\log {\big [}\mathrm {H} _{3}\mathrm {O} ^{+}{\big ]}}"></span>. </p><p>Le pH de l'eau pure est de 7, ce qui signifie qu'il y 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 10^{-7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>7</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f7bc29238b21553311d8359274b4782ecb3f73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.658ex; height:2.676ex;" alt="{\displaystyle 10^{-7}}"></span> mole 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 \mathrm {H_{3}} \mathrm {O} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H_{3}} \mathrm {O} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f5fe782f551c1c3ef7f1aa32f7cafbd988a8fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.116ex; height:2.843ex;" alt="{\displaystyle \mathrm {H_{3}} \mathrm {O} ^{+}}"></span> dans un litre d'eau. </p><p>Le pH du jus de citron est de 2,4, ce qui signifie qu'il y 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 10^{-2,4}=4\cdot 10^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{-2,4}=4\cdot 10^{-3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe179368808c79b37697167ab2491352e96ac14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.535ex; height:2.676ex;" alt="{\displaystyle 10^{-2,4}=4\cdot 10^{-3}}"></span> mole 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 \mathrm {H_{3}} \mathrm {O} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H_{3}} \mathrm {O} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f5fe782f551c1c3ef7f1aa32f7cafbd988a8fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.116ex; height:2.843ex;" alt="{\displaystyle \mathrm {H_{3}} \mathrm {O} ^{+}}"></span> dans un litre de jus de citron. </p><p>On remarque qu'un pH faible correspond à une concentration élevée 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 \mathrm {H} _{3}\mathrm {O} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {H} _{3}\mathrm {O} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4c17daddf405eaf4b7ed150bf1cf63ece22c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.116ex; height:2.843ex;" alt="{\displaystyle \mathrm {H} _{3}\mathrm {O} ^{+}}"></span> donc à un milieu acide. </p> <div class="mw-heading mw-heading3"><h3 id="Les_décibels"><span id="Les_d.C3.A9cibels"></span>Les décibels</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=8" title="Modifier la section : Les décibels" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&action=edit&section=8" title="Modifier le code source de la section : Les décibels"><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/D%C3%A9cibel" title="Décibel">Décibel</a>.</div></div> <p>En acoustique, une différence d'un décibel (dB) entre deux puissances signifie que le logarithme du rapport entre ces deux puissances est de 0,1 (un dixième de <a href="/wiki/Bel" title="Bel">bel</a>). Sachant qu'un logarithme de 0,1 correspond à un nombre égal à 1,26, une augmentation de 1 dB correspond à une multiplication de la puissance par 1,26. Une multiplication de la puissance sonore par 2 correspond à une augmentation de 3 dB car <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 10^{0,3}\approx 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo>≈</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{0,3}\approx 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab44b42b758aa80db50eef2302d43d6d54e2d53c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.919ex; height:2.676ex;" alt="{\displaystyle 10^{0,3}\approx 2}"></span>. </p><p>Mathématiquement : soit β le niveau sonore : <b>β = I(dB) = 10 log(I/Ii)</b> où I est l'intensité sonore et Ii l'intensité de référence. </p><p>La variation <b>Δβ</b> sera donc égale au logarithme décimal du rapport des intensités I1 et I2 (Δβ = 10 log(I1/I2)), et ceci grâce à la propriété des logarithmes décimaux : log(<i>a</i>)−log(<i>b</i>) = log(<i>a</i>/<i>b</i>). </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_d%C3%A9cimal&veaction=edit&section=9" 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_d%C3%A9cimal&action=edit&section=9" 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> <div class="references-small decimal" style=""><div class="mw-references-wrap"><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"><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-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="HardyWright"><span class="ouvrage" id="G.H._HardyE.M._Wright"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> G.H. Hardy et E.M. Wright, <cite class="italique" lang="en">An Introduction to the Theory of Numbers</cite>, Oxford, Clarendon Press <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="//books.google.com/books?id=FlUj0Rk_rF4C&pg=PA8">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 8<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.place=Oxford&rft.pub=Clarendon+Press&rft.aulast=Hardy&rft.aufirst=G.H.&rft.au=E.M.+Wright&rft.pages=8&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ALogarithme+d%C3%A9cimal"></span></span></span>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Articles_connexes">Articles connexes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logarithme_d%C3%A9cimal&veaction=edit&section=10" 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_d%C3%A9cimal&action=edit&section=10" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Logarithme" title="Logarithme">Logarithme</a></li> <li><a href="/wiki/Logarithme_binaire" title="Logarithme binaire">Logarithme binaire</a></li> <li><a href="/wiki/Logarithme_n%C3%A9p%C3%A9rien" title="Logarithme népérien">Logarithme néperien</a></li> <li><a href="/wiki/Table_de_logarithmes" title="Table de logarithmes">Table de logarithmes</a></li></ul> <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" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/36px-Nuvola_apps_kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Nuvola_apps_kmplot.svg/48px-Nuvola_apps_kmplot.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Analyse" title="Portail:Analyse">Portail de l'analyse</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Physique" title="Portail de la physique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/36px-Circle-icons-physics-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/48px-Circle-icons-physics-logo.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Physique" title="Portail:Physique">Portail de la physique</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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