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Naturliga logaritmen – Wikipedia
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href="#Derivata_och_taylorserier"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Derivata och taylorserier</span> </div> </a> <ul id="toc-Derivata_och_taylorserier-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referenser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referenser"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Referenser</span> </div> </a> <button aria-controls="toc-Referenser-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>Växla underavsnittet Referenser</span> </button> <ul id="toc-Referenser-sublist" class="vector-toc-list"> <li id="toc-Noter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noter"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Noter</span> 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class="vector-dropdown-label-text">Växla innehållsförteckningen</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">Naturliga logaritmen</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="Gå till en artikel på ett annat språk. Tillgänglig på 61 språk" > <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-61" 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">61 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Natuurlike_logaritme" title="Natuurlike logaritme – afrikaans" lang="af" hreflang="af" data-title="Natuurlike logaritme" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D9%88%D8%BA%D8%A7%D8%B1%D9%8A%D8%AA%D9%85_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A" title="لوغاريتم طبيعي – arabiska" lang="ar" hreflang="ar" data-title="لوغاريتم طبيعي" data-language-autonym="العربية" data-language-local-name="arabiska" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Natural_loqarifm" title="Natural loqarifm – azerbajdzjanska" lang="az" hreflang="az" data-title="Natural loqarifm" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzjanska" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C5%AB-ji%C3%A2n_t%C3%B9i-s%C3%B2%CD%98" title="Chū-jiân tùi-sò͘ – min nan" lang="nan" hreflang="nan" data-title="Chū-jiân tùi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B_%D0%BB%D0%B0%D0%B3%D0%B0%D1%80%D1%8B%D1%84%D0%BC" title="Натуральны лагарыфм – belarusiska" lang="be" hreflang="be" data-title="Натуральны лагарыфм" data-language-autonym="Беларуская" data-language-local-name="belarusiska" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D0%B5%D0%BD_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D1%8A%D0%BC" title="Натурален логаритъм – bulgariska" lang="bg" hreflang="bg" data-title="Натурален логаритъм" data-language-autonym="Български" data-language-local-name="bulgariska" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prirodni_logaritam" title="Prirodni logaritam – bosniska" lang="bs" hreflang="bs" data-title="Prirodni logaritam" data-language-autonym="Bosanski" data-language-local-name="bosniska" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Logaritm_neperian" title="Logaritm neperian – bretonska" lang="br" hreflang="br" data-title="Logaritm neperian" data-language-autonym="Brezhoneg" data-language-local-name="bretonska" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Logaritme_natural" title="Logaritme natural – katalanska" lang="ca" hreflang="ca" data-title="Logaritme natural" data-language-autonym="Català" data-language-local-name="katalanska" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D0%BB%C4%83_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Натураллă логарифм – tjuvasjiska" lang="cv" hreflang="cv" data-title="Натураллă логарифм" data-language-autonym="Чӑвашла" data-language-local-name="tjuvasjiska" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Naturlig_logaritme" title="Naturlig logaritme – danska" lang="da" hreflang="da" data-title="Naturlig logaritme" data-language-autonym="Dansk" data-language-local-name="danska" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Nat%C3%BCrlicher_Logarithmus" title="Natürlicher Logarithmus – tyska" lang="de" hreflang="de" data-title="Natürlicher Logarithmus" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Naturaallogaritm" title="Naturaallogaritm – estniska" lang="et" hreflang="et" data-title="Naturaallogaritm" data-language-autonym="Eesti" data-language-local-name="estniska" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A6%CF%85%CF%83%CE%B9%CE%BA%CF%8C%CF%82_%CE%BB%CE%BF%CE%B3%CE%AC%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82" title="Φυσικός λογάριθμος – grekiska" lang="el" hreflang="el" data-title="Φυσικός λογάριθμος" data-language-autonym="Ελληνικά" data-language-local-name="grekiska" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Natural_logarithm" title="Natural logarithm – engelska" lang="en" hreflang="en" data-title="Natural logarithm" data-language-autonym="English" data-language-local-name="engelska" 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_natural" title="Logaritmo natural – spanska" lang="es" hreflang="es" data-title="Logaritmo natural" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Logaritmo_natural" title="Logaritmo natural – baskiska" lang="eu" hreflang="eu" data-title="Logaritmo natural" data-language-autonym="Euskara" data-language-local-name="baskiska" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%84%DA%AF%D8%A7%D8%B1%DB%8C%D8%AA%D9%85_%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C" title="لگاریتم طبیعی – persiska" lang="fa" hreflang="fa" data-title="لگاریتم طبیعی" data-language-autonym="فارسی" data-language-local-name="persiska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/N%C3%A1tt%C3%BArlig_logaritma" title="Náttúrlig logaritma – färöiska" lang="fo" hreflang="fo" data-title="Náttúrlig logaritma" data-language-autonym="Føroyskt" data-language-local-name="färöiska" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Logarithme_n%C3%A9p%C3%A9rien" title="Logarithme népérien – franska" lang="fr" hreflang="fr" data-title="Logarithme népérien" data-language-autonym="Français" data-language-local-name="franska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Logaritmo_natural" title="Logaritmo natural – galiciska" lang="gl" hreflang="gl" data-title="Logaritmo natural" data-language-autonym="Galego" data-language-local-name="galiciska" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EC%97%B0%EB%A1%9C%EA%B7%B8" title="자연로그 – koreanska" lang="ko" hreflang="ko" data-title="자연로그" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6_%D5%AC%D5%B8%D5%A3%D5%A1%D6%80%D5%AB%D5%A9%D5%B4" title="Բնական լոգարիթմ – armeniska" lang="hy" hreflang="hy" data-title="Բնական լոգարիթմ" data-language-autonym="Հայերեն" data-language-local-name="armeniska" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%95%E0%A5%83%E0%A4%A4%E0%A4%BF%E0%A4%95_%E0%A4%B2%E0%A4%98%E0%A5%81%E0%A4%97%E0%A4%A3%E0%A4%95" title="प्राकृतिक लघुगणक – hindi" lang="hi" hreflang="hi" data-title="प्राकृतिक लघुगणक" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logaritma_alami" title="Logaritma alami – indonesiska" lang="id" hreflang="id" data-title="Logaritma alami" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiska" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logaritmo_naturale" title="Logaritmo naturale – italienska" lang="it" hreflang="it" data-title="Logaritmo naturale" data-language-autonym="Italiano" data-language-local-name="italienska" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9C%D7%95%D7%92%D7%A8%D7%99%D7%AA%D7%9D_%D7%98%D7%91%D7%A2%D7%99" title="לוגריתם טבעי – hebreiska" lang="he" hreflang="he" data-title="לוגריתם טבעי" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9C%E1%83%90%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%90%E1%83%9A%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9A%E1%83%9D%E1%83%92%E1%83%90%E1%83%A0%E1%83%98%E1%83%97%E1%83%9B%E1%83%98" title="ნატურალური ლოგარითმი – georgiska" lang="ka" hreflang="ka" data-title="ნატურალური ლოგარითმი" data-language-autonym="ქართული" data-language-local-name="georgiska" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Натурал логарифм – kazakiska" lang="kk" hreflang="kk" data-title="Натурал логарифм" data-language-autonym="Қазақша" data-language-local-name="kazakiska" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Natur%C4%81lais_logaritms" title="Naturālais logaritms – lettiska" lang="lv" hreflang="lv" data-title="Naturālais logaritms" data-language-autonym="Latviešu" data-language-local-name="lettiska" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Nat%C5%ABrinis_logaritmas" title="Natūrinis logaritmas – litauiska" lang="lt" hreflang="lt" data-title="Natūrinis logaritmas" data-language-autonym="Lietuvių" data-language-local-name="litauiska" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Logaritm_natural" title="Logaritm natural – lombardiska" lang="lmo" hreflang="lmo" data-title="Logaritm natural" data-language-autonym="Lombard" data-language-local-name="lombardiska" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Term%C3%A9szetes_logaritmus" title="Természetes logaritmus – ungerska" lang="hu" hreflang="hu" data-title="Természetes logaritmus" data-language-autonym="Magyar" data-language-local-name="ungerska" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%B5%D0%BD_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Природен логаритам – makedonska" lang="mk" hreflang="mk" data-title="Природен логаритам" data-language-autonym="Македонски" data-language-local-name="makedonska" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logaritma_asli" title="Logaritma asli – malajiska" lang="ms" hreflang="ms" data-title="Logaritma asli" data-language-autonym="Bahasa Melayu" data-language-local-name="malajiska" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Natuurlijke_logaritme" title="Natuurlijke logaritme – nederländska" lang="nl" hreflang="nl" data-title="Natuurlijke logaritme" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%AF%BE%E6%95%B0" title="自然対数 – japanska" lang="ja" hreflang="ja" data-title="自然対数" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Naturlig_logaritme" title="Naturlig logaritme – norskt bokmål" lang="nb" hreflang="nb" data-title="Naturlig logaritme" data-language-autonym="Norsk bokmål" data-language-local-name="norskt bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Logaritme_neperian" title="Logaritme neperian – occitanska" lang="oc" hreflang="oc" data-title="Logaritme neperian" data-language-autonym="Occitan" data-language-local-name="occitanska" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Logaritm_%C3%ABd_Napier" title="Logaritm ëd Napier – piemontesiska" lang="pms" hreflang="pms" data-title="Logaritm ëd Napier" data-language-autonym="Piemontèis" data-language-local-name="piemontesiska" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logarytm_naturalny" title="Logarytm naturalny – polska" lang="pl" hreflang="pl" data-title="Logarytm naturalny" data-language-autonym="Polski" data-language-local-name="polska" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Logaritmo_natural" title="Logaritmo natural – portugisiska" lang="pt" hreflang="pt" data-title="Logaritmo natural" data-language-autonym="Português" data-language-local-name="portugisiska" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Logaritm_natural" title="Logaritm natural – rumänska" lang="ro" hreflang="ro" data-title="Logaritm natural" data-language-autonym="Română" data-language-local-name="rumänska" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Натуральный логарифм – ryska" lang="ru" hreflang="ru" data-title="Натуральный логарифм" data-language-autonym="Русский" data-language-local-name="ryska" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logaritmi_natyror" title="Logaritmi natyror – albanska" lang="sq" hreflang="sq" data-title="Logaritmi natyror" data-language-autonym="Shqip" data-language-local-name="albanska" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%9A%E0%B7%98%E0%B6%AD%E0%B7%92_%E0%B6%BD%E0%B6%9D%E0%B7%94%E0%B6%9C%E0%B6%AB%E0%B6%9A" title="ප්රකෘති ලඝුගණක – singalesiska" lang="si" hreflang="si" data-title="ප්රකෘති ලඝුගණක" data-language-autonym="සිංහල" data-language-local-name="singalesiska" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Naravni_logaritem" title="Naravni logaritem – slovenska" lang="sl" hreflang="sl" data-title="Naravni logaritem" data-language-autonym="Slovenščina" data-language-local-name="slovenska" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%86%DA%AF%D8%A7%D8%B1%DB%8C%D8%AA%D9%85%DB%8C_%D8%B3%D8%B1%D9%88%D8%B4%D8%AA%DB%8C" title="لۆگاریتمی سروشتی – centralkurdiska" lang="ckb" hreflang="ckb" data-title="لۆگاریتمی سروشتی" data-language-autonym="کوردی" data-language-local-name="centralkurdiska" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%BD%D0%B8_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Природни логаритам – serbiska" lang="sr" hreflang="sr" data-title="Природни логаритам" data-language-autonym="Српски / srpski" data-language-local-name="serbiska" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Prirodni_logaritam" title="Prirodni logaritam – serbokroatiska" lang="sh" hreflang="sh" data-title="Prirodni logaritam" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatiska" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Luonnollinen_logaritmi" title="Luonnollinen logaritmi – finska" lang="fi" hreflang="fi" data-title="Luonnollinen logaritmi" data-language-autonym="Suomi" data-language-local-name="finska" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%AE%E0%AE%9F%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%88" title="இயல் மடக்கை – tamil" lang="ta" hreflang="ta" data-title="இயல் மடக்கை" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Alugaritm_agaman" title="Alugaritm agaman – tachelhit" lang="shi" hreflang="shi" data-title="Alugaritm agaman" data-language-autonym="Taclḥit" data-language-local-name="tachelhit" class="interlanguage-link-target"><span>Taclḥit</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%AD%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%B4%E0%B8%97%E0%B8%B6%E0%B8%A1%E0%B8%98%E0%B8%A3%E0%B8%A3%E0%B8%A1%E0%B8%8A%E0%B8%B2%E0%B8%95%E0%B8%B4" title="ลอการิทึมธรรมชาติ – thailändska" lang="th" hreflang="th" data-title="ลอการิทึมธรรมชาติ" data-language-autonym="ไทย" data-language-local-name="thailändska" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BB%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Натуральний логарифм – ukrainska" lang="uk" hreflang="uk" data-title="Натуральний логарифм" data-language-autonym="Українська" data-language-local-name="ukrainska" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%D8%AF%D8%B1%D8%AA%DB%8C_%D9%84%D8%A7%DA%AF%D8%B1%D8%AA%DA%BE%D9%85" title="قدرتی لاگرتھم – urdu" lang="ur" hreflang="ur" data-title="قدرتی لاگرتھم" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Logarit_t%E1%BB%B1_nhi%C3%AAn" title="Logarit tự nhiên – vietnamesiska" lang="vi" hreflang="vi" data-title="Logarit tự nhiên" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesiska" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%B0%8D%E6%95%B8" title="自然對數 – litterär kineiska" lang="lzh" hreflang="lzh" data-title="自然對數" data-language-autonym="文言" data-language-local-name="litterär kineiska" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%AF%B9%E6%95%B0" title="自然对数 – wu" lang="wuu" hreflang="wuu" data-title="自然对数" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%B0%8D%E6%95%B8" title="自然對數 – kantonesiska" lang="yue" hreflang="yue" data-title="自然對數" data-language-autonym="粵語" data-language-local-name="kantonesiska" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%B0%8D%E6%95%B8" title="自然對數 – kinesiska" lang="zh" hreflang="zh" data-title="自然對數" data-language-autonym="中文" data-language-local-name="kinesiska" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q204037#sitelinks-wikipedia" title="Redigera interwikilänkar" class="wbc-editpage">Redigera länkar</a></span></div> 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Från Wikipedia</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="sv" dir="ltr"><div class="noexcerpt noprint hanvisning_bas"> <dl><dd><i>Uppslagsordet ”ln” leder hit.  För Unix-kommandot, se <a href="/wiki/Ln_(kommando)" title="Ln (kommando)">ln (kommando)</a>.  </i></dd></dl></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Naturliga-logaritmen.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Naturliga-logaritmen.png/250px-Naturliga-logaritmen.png" decoding="async" width="250" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Naturliga-logaritmen.png/375px-Naturliga-logaritmen.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Naturliga-logaritmen.png/500px-Naturliga-logaritmen.png 2x" data-file-width="800" data-file-height="683" /></a><figcaption>Naturliga logaritmen</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:LogaritmusNaturalis.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/LogaritmusNaturalis.png/250px-LogaritmusNaturalis.png" decoding="async" width="250" height="245" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/LogaritmusNaturalis.png/375px-LogaritmusNaturalis.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/LogaritmusNaturalis.png/500px-LogaritmusNaturalis.png 2x" data-file-width="800" data-file-height="785" /></a><figcaption><a href="/wiki/Hyperbel" title="Hyperbel">Hyperbeln</a> <i>y</i> = 1/<i>x</i> (blå kurva) och arean från <i>x</i> = 1 till 6 (skuggad). Denna area är lika med den naturliga logaritmen av 6.</figcaption></figure> <p><b>Naturliga logaritmen</b> är en <a href="/wiki/Logaritm" title="Logaritm">logaritm</a> med basen <i><a href="/wiki/E_(tal)" title="E (tal)">e</a></i>, ett <a href="/wiki/Transcendenta_tal" title="Transcendenta tal">transcendent</a> tal approximativt lika med 2,718. Den naturliga logaritmen av ett tal <i>x</i> skrivs ofta ln(<i>x</i>) och är <a href="/wiki/Definitionsm%C3%A4ngd" title="Definitionsmängd">definierad</a> för alla strikt <a href="/wiki/Positiva_tal" title="Positiva tal">positiva tal</a>.<sup id="cite_ref-Mortimer2005_1-0" class="reference"><a href="#cite_note-Mortimer2005-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup> Den naturliga logaritmfunktionen är en reellvärd funktion av en reell variabel: </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} ^{\ln x}=x\qquad {\mbox{om }}x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>om </mtext> </mstyle> </mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {e} ^{\ln x}=x\qquad {\mbox{om }}x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea361b93f6bfa5f65f49476737904e2b61dfb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:22.306ex; height:2.676ex;" alt="{\displaystyle \mathrm {e} ^{\ln x}=x\qquad {\mbox{om }}x>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 \ln \mathrm {e} ^{x}=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \mathrm {e} ^{x}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb549cdb5fad1265c6cd2405aea575220e004f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.959ex; height:2.343ex;" alt="{\displaystyle \ln \mathrm {e} ^{x}=x}"></span></dd></dl> <p>I likhet med alla logaritmiska funktioner, mappas <a href="/wiki/Multiplikation" title="Multiplikation">multiplikation</a> till addition: </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(xy)=\ln x+\ln y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(xy)=\ln x+\ln y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6461afe36aa9a8f3b995e83f2042ce9c7da31a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.311ex; height:2.843ex;" alt="{\displaystyle \ln(xy)=\ln x+\ln y}"></span></dd></dl> <p>Naturliga logaritmen kan definieras med integralen </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=\int _{1}^{x}{\frac {1}{t}}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e72c70a86d7ec8c9b4353058bda339ff8598c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.832ex; height:5.843ex;" alt="{\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}dt}"></span></dd></dl> <p>Ett tidigt omnämnande av <i>naturlig logaritm</i> gjordes av <a href="/wiki/Nicholas_Mercator" class="mw-redirect" title="Nicholas Mercator">Nicholas Mercator</a> i verket <i>Logarithmotechnia</i> 1658, men matematikläraren John Speidell hade redan 1619 sammanställt en tabell över naturliga logaritmer.<sup id="cite_ref-Connor2001_2-0" class="reference"><a href="#cite_note-Connor2001-2"><span class="cite-reference-link-bracket">[</span>2<span class="cite-reference-link-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Egenskaper">Egenskaper</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naturliga_logaritmen&veaction=edit&section=1" title="Redigera avsnitt: Egenskaper" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Naturliga_logaritmen&action=edit&section=1" title="Redigera avsnitts källkod: Egenskaper"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln \mathrm {e} =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln \mathrm {e} =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6184071f5ccf9165e8903d319f26eab8653a68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.62ex; height:2.176ex;" alt="{\displaystyle \ln \mathrm {e} =1}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(xy)=\ln x+\ln y;\quad \quad x>0,\ y>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> <mo>+</mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>y</mi> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(xy)=\ln x+\ln y;\quad \quad x>0,\ y>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a761fa889646c78057b185d68d7a413b21472c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.611ex; height:2.843ex;" alt="{\displaystyle \ln(xy)=\ln x+\ln y;\quad \quad x>0,\ y>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\frac {x}{y}}=\ln x-\ln y;\quad \quad x>0,\ y>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> <mo>−<!-- − --></mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>y</mi> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>y</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln {\frac {x}{y}}=\ln x-\ln y;\quad \quad x>0,\ y>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9efe72adf93a529e803d64301d2e925a5344cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.87ex; height:5.176ex;" alt="{\displaystyle \ln {\frac {x}{y}}=\ln x-\ln y;\quad \quad x>0,\ y>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x<\ln y;\quad \quad 0<x<y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> <mo><</mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>y</mi> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x<\ln y;\quad \quad 0<x<y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71573697a48520dfad06a6e2b7167541392f4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.76ex; height:2.509ex;" alt="{\displaystyle \ln x<\ln y;\quad \quad 0<x<y}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a32ef3d5fbd5df0406c68a2ad58e415ae641d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.971ex; height:5.843ex;" alt="{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to 0}{\frac {x^{n}-1}{n}}=\ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to 0}{\frac {x^{n}-1}{n}}=\ln x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f112e31860df1a3c1bcb7b269da6cdcd8c66f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.98ex; height:5.343ex;" alt="{\displaystyle \lim _{n\to 0}{\frac {x^{n}-1}{n}}=\ln x}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x^{y}=y\,\ln x;\quad \quad x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mi>y</mi> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x^{y}=y\,\ln x;\quad \quad x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f1210a7dbacbe202dae76a1c20677e8ec7c73d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.659ex; height:2.676ex;" alt="{\displaystyle \ln x^{y}=y\,\ln x;\quad \quad x>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1;\quad \quad x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>≤<!-- ≤ --></mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mi>x</mi> <mo>≤<!-- ≤ --></mo> <mi>x</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1;\quad \quad x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf10296a2a574caa346f6ee0d9bd62bf3c6966e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.624ex; height:5.176ex;" alt="{\displaystyle {\frac {x-1}{x}}\leq \ln x\leq x-1;\quad \quad x>0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x;\quad \quad x\geq 0,\ \alpha \geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>≤<!-- ≤ --></mo> <mi>α<!-- α --></mi> <mi>x</mi> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>x</mi> <mo>≥<!-- ≥ --></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>α<!-- α --></mi> <mo>≥<!-- ≥ --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x;\quad \quad x\geq 0,\ \alpha \geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6451397e7e77b9eeb9b6cd625cb8eef3f15622f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.301ex; height:2.843ex;" alt="{\displaystyle \ln {(1+x^{\alpha })}\leq \alpha x;\quad \quad x\geq 0,\ \alpha \geq 1}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Derivata_och_taylorserier">Derivata och taylorserier</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naturliga_logaritmen&veaction=edit&section=2" title="Redigera avsnitt: Derivata och taylorserier" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Naturliga_logaritmen&action=edit&section=2" title="Redigera avsnitts källkod: Derivata och taylorserier"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:LogTay.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/LogTay.svg/300px-LogTay.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/LogTay.svg/450px-LogTay.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/LogTay.svg/600px-LogTay.svg.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>Taylorpolynomen för ln(1 + <i>x</i>) ger noggranna approximationer endast i intervallet −1 < <i>x</i> ≤ 1. Notera att, för <i>x</i> > 1, ger taylorpolynomen av högre gradtal <i>sämre</i> approximationer</figcaption></figure> <p>Den naturliga logaritmens <a href="/wiki/Derivata" title="Derivata">derivata</a> ges av </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 {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e0f2d4a2ba2eaa4aaea0752486212d65af14c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.188ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}}"></span></dd></dl> <p>Bevis: </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 {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)=\lim _{h\to 0}{\frac {\ln(x+h)-\ln(x)}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)=\lim _{h\to 0}{\frac {\ln(x+h)-\ln(x)}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b664a8ed3f0b0b3a00d9bf726e1263b27006b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.834ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)=\lim _{h\to 0}{\frac {\ln(x+h)-\ln(x)}{h}}}"></span> <dl><dd><dl><dd><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 =\lim _{h\to 0}{\frac {\ln({\frac {x+h}{x}})}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>h</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\lim _{h\to 0}{\frac {\ln({\frac {x+h}{x}})}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b08aae24f97c4f91c2011a049bb0ab05c970041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.839ex; height:6.676ex;" alt="{\displaystyle =\lim _{h\to 0}{\frac {\ln({\frac {x+h}{x}})}{h}}}"></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 =\lim _{h\to 0}\left[{\frac {1}{h}}\ln \left(1+{\frac {h}{x}}\right)\right]\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\lim _{h\to 0}\left[{\frac {1}{h}}\ln \left(1+{\frac {h}{x}}\right)\right]\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d8ec064c03dca33ec9440c431407ea9090bbe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.131ex; height:6.176ex;" alt="{\displaystyle =\lim _{h\to 0}\left[{\frac {1}{h}}\ln \left(1+{\frac {h}{x}}\right)\right]\quad }"></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 =\lim _{h\to 0}\ln \left(\left[1+{\frac {h}{x}}\right]^{\frac {1}{h}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>x</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\lim _{h\to 0}\ln \left(\left[1+{\frac {h}{x}}\right]^{\frac {1}{h}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1d3421de5d990d1e29ca4b0fafe6af365381cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:22.729ex; height:8.176ex;" alt="{\displaystyle =\lim _{h\to 0}\ln \left(\left[1+{\frac {h}{x}}\right]^{\frac {1}{h}}\right)}"></span></dd></dl></dd></dl></dd></dl></dd> <dd>Låt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {h}{x}}\Rightarrow ux=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mi>x</mi> </mfrac> </mrow> <mo stretchy="false">⇒<!-- ⇒ --></mo> <mi>u</mi> <mi>x</mi> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {h}{x}}\Rightarrow ux=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/320065355f1d23bd07343e77b588ee450252ac2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.314ex; height:5.343ex;" alt="{\displaystyle u={\frac {h}{x}}\Rightarrow ux=h}"></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 {\frac {1}{h}}={\frac {1}{ux}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>u</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{h}}={\frac {1}{ux}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d19b2c0f15b25a082e5d381d002dfa2c72f38efa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.769ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{h}}={\frac {1}{ux}}}"></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 {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)=\lim _{u\to 0}\ln \left([1+u]^{\frac {1}{ux}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>u</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)=\lim _{u\to 0}\ln \left([1+u]^{\frac {1}{ux}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d76ecd656eb1c9bc2f4b0c36510b898d9118c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.751ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)=\lim _{u\to 0}\ln \left([1+u]^{\frac {1}{ux}}\right)}"></span> <dl><dd><dl><dd><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 =\lim _{u\to 0}\ln \left(\left[[1+u]^{\frac {1}{u}}\right]^{\frac {1}{x}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <msup> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\lim _{u\to 0}\ln \left(\left[[1+u]^{\frac {1}{u}}\right]^{\frac {1}{x}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc6b043305c66967e5cf62264a3a185c7590c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.351ex; height:7.509ex;" alt="{\displaystyle =\lim _{u\to 0}\ln \left(\left[[1+u]^{\frac {1}{u}}\right]^{\frac {1}{x}}\right)}"></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 ={\frac {1}{x}}\lim _{u\to 0}\ln \left([1+u]^{\frac {1}{u}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo stretchy="false">→<!-- → --></mo> <mn>0</mn> </mrow> </munder> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{x}}\lim _{u\to 0}\ln \left([1+u]^{\frac {1}{u}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b35452d8970ca852763123618f645184870fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.971ex; height:5.343ex;" alt="{\displaystyle ={\frac {1}{x}}\lim _{u\to 0}\ln \left([1+u]^{\frac {1}{u}}\right)}"></span></dd></dl></dd></dl></dd></dl></dd> <dd>Låt <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={\frac {1}{u}}\Rightarrow u={\frac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> <mo stretchy="false">⇒<!-- ⇒ --></mo> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {1}{u}}\Rightarrow u={\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0f27fd2ec184f1bf47cdae77799a887f58f4be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.932ex; height:5.176ex;" alt="{\displaystyle n={\frac {1}{u}}\Rightarrow u={\frac {1}{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 {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}\lim _{n\to \infty }\ln \left(1+{\frac {1}{n}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <mi>ln</mi> <mo>⁡<!-- --></mo> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}\lim _{n\to \infty }\ln \left(1+{\frac {1}{n}}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9113cc230fa8be18ffeb02dc3a95528878ede89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.435ex; height:6.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\ln(x)={\frac {1}{x}}\lim _{n\to \infty }\ln \left(1+{\frac {1}{n}}\right)^{n}}"></span> <dl><dd><dl><dd><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 ={\frac {1}{x}}\ln \left(\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow> <mo>(</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→<!-- → --></mo> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{x}}\ln \left(\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc6531a6aa311bcf5aa7ed1c2045de61e659d58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.9ex; height:6.176ex;" alt="{\displaystyle ={\frac {1}{x}}\ln \left(\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}\right)}"></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 ={\frac {1}{x}}\ln \mathrm {e} \ =\ {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡<!-- --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{x}}\ln \mathrm {e} \ =\ {\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0819cb8d584d2e2598a2803f732e7f6e4c44a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.791ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{x}}\ln \mathrm {e} \ =\ {\frac {1}{x}}}"></span></dd></dl></dd></dl></dd></dl></dd></dl> <p>Detta leder till <a href="/wiki/Taylorserie" title="Taylorserie">taylorserierna</a> för ln(1 + <i>x</i>) kring 0 (också kända som <i>mercatorserierna</i>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots \ ;\qquad \left|x\right|\leq 1,\ x\neq -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡<!-- --></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−<!-- − --></mo> <mo>⋯<!-- ⋯ --></mo> <mtext> </mtext> <mo>;</mo> <mspace width="2em" /> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>≤<!-- ≤ --></mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mi>x</mi> <mo>≠<!-- ≠ --></mo> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots \ ;\qquad \left|x\right|\leq 1,\ x\neq -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37c950ccdca4e9abfeea37e3fe2de25009fdac68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:72.062ex; height:7.009ex;" alt="{\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots \ ;\qquad \left|x\right|\leq 1,\ x\neq -1}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Referenser">Referenser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naturliga_logaritmen&veaction=edit&section=3" title="Redigera avsnitt: Referenser" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Naturliga_logaritmen&action=edit&section=3" title="Redigera avsnitts källkod: Referenser"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Noter">Noter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Naturliga_logaritmen&veaction=edit&section=4" title="Redigera avsnitt: Noter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Naturliga_logaritmen&action=edit&section=4" title="Redigera avsnitts källkod: Noter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Mortimer2005-1"><a href="#cite_ref-Mortimer2005_1-0">^</a> <span class="reference-text"><cite style="font-style:normal" class="book" id="CITEREFMortimer2005">Mortimer, Robert G. (2005). <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGoSv5tmATsC">Mathematics for physical chemistry</a></i> (3rd). Academic Press. sid. 9. <a href="/wiki/Special:Bokk%C3%A4llor/0-12-508347-5" title="Special:Bokkällor/0-12-508347-5">ISBN 0-12-508347-5</a><span class="printonly">. <a rel="nofollow" class="external free" href="https://books.google.com/books?id=nGoSv5tmATsC">https://books.google.com/books?id=nGoSv5tmATsC</a></span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+physical+chemistry&rft.aulast=Mortimer&rft.aufirst=Robert+G.&rft.au=Mortimer%2C+Robert+G.&rft.date=2005&rft.pages=sid.%26nbsp%3B9&rft.edition=3rd&rft.pub=Academic+Press&rft.isbn=0-12-508347-5&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnGoSv5tmATsC&rfr_id=info:sid/en.wikipedia.org:Naturliga_logaritmen"><span style="display: none;"> </span></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nGoSv5tmATsC&pg=PA9">Extract of page 9</a></span> </li> <li id="cite_note-Connor2001-2"><a href="#cite_ref-Connor2001_2-0">^</a> <span class="reference-text"><cite style="font-style:normal" class="web" id="CITEREFJ_J_O'Connor_and_E_F_Robertson2001">J J O'Connor and E F Robertson (1 september 2001). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/HistTopics/e.html">”The number e”</a>. The MacTutor History of Mathematics archive<span class="printonly">. <a rel="nofollow" class="external free" href="http://www-history.mcs.st-and.ac.uk/HistTopics/e.html">http://www-history.mcs.st-and.ac.uk/HistTopics/e.html</a></span><span class="reference-accessdate">. Läst 2 februari 2009</span>.</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=The+number+e&rft.atitle=&rft.aulast=J+J+O%27Connor+and+E+F+Robertson&rft.au=J+J+O%27Connor+and+E+F+Robertson&rft.date=1+september+2001&rft.pub=The+MacTutor+History+of+Mathematics+archive&rft_id=http%3A%2F%2Fwww-history.mcs.st-and.ac.uk%2FHistTopics%2Fe.html&rfr_id=info:sid/en.wikipedia.org:Naturliga_logaritmen"><span style="display: none;"> </span></span></span> </li> </ol></div> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐5dc468848‐g7zhs Cached time: 20241122180954 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.128 seconds Real time usage: 0.624 seconds Preprocessor visited node count: 1671/1000000 Post‐expand include size: 9842/2097152 bytes Template argument size: 3309/2097152 bytes Highest expansion depth: 18/100 Expensive parser function count: 0/500 Unstrip recursion depth: 0/20 Unstrip post‐expand size: 4227/5000000 bytes Lua time usage: 0.007/10.000 seconds Lua memory usage: 748091/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 90.070 1 -total 60.66% 54.638 2 Mall:Citation/core 48.05% 43.277 1 Mall:Bokref 44.57% 40.140 1 Mall:Cite_book 36.98% 33.312 1 Mall:Webbref 24.21% 21.804 1 Mall:Cite_web 11.59% 10.436 3 Mall:Date 10.73% 9.666 1 Mall:Omdirigering 6.39% 5.752 1 Mall:Hänvisning_bas 6.10% 5.495 1 Mall:ISBN --> <!-- Saved in parser cache with key svwiki:pcache:idhash:37985-0!canonical and timestamp 20241122180954 and revision id 50529764. 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