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class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_bs.wikipedia.org&uselang=bs" class=""><span>Donacije</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Posebno:Napravi_ra%C4%8Dun&returnto=Logaritam" title="Predlažemo vam da napravite račun i prijavite se; međutim, to nije obavezno" class=""><span>Napravi korisnički račun</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Posebno:Korisni%C4%8Dka_prijava&returnto=Logaritam" title="Predlažemo vam da se prijavite, iako to nije obavezno [o]" accesskey="o" class=""><span>Prijavi 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id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Početak</div> </a> </li> <li id="toc-Motivacija_i_definicija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Motivacija_i_definicija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Motivacija i definicija</span> </div> </a> <button aria-controls="toc-Motivacija_i_definicija-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>Uključi/isključi podsekciju Motivacija i definicija</span> </button> <ul id="toc-Motivacija_i_definicija-sublist" class="vector-toc-list"> <li id="toc-Eksponencija" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eksponencija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Eksponencija</span> </div> </a> <ul id="toc-Eksponencija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definicija" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definicija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definicija</span> </div> </a> <ul id="toc-Definicija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primjeri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primjeri"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Primjeri</span> </div> </a> <ul id="toc-Primjeri-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Logaritamski_identiteti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Logaritamski_identiteti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Logaritamski identiteti</span> </div> </a> <button aria-controls="toc-Logaritamski_identiteti-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>Uključi/isključi podsekciju Logaritamski identiteti</span> </button> <ul id="toc-Logaritamski_identiteti-sublist" class="vector-toc-list"> <li id="toc-Proizvod,_koeficijent,_stepen_i_korijen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proizvod,_koeficijent,_stepen_i_korijen"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Proizvod, koeficijent, stepen i korijen</span> </div> </a> <ul id="toc-Proizvod,_koeficijent,_stepen_i_korijen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Izmjena_baze" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Izmjena_baze"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Izmjena baze</span> </div> </a> <ul id="toc-Izmjena_baze-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Određene_baze" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Određene_baze"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Određene baze</span> </div> </a> <ul id="toc-Određene_baze-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historija"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Historija</span> </div> </a> <ul id="toc-Historija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logaritamke_tablice,_logaritamska_skala_i_historijske_primjene" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Logaritamke_tablice,_logaritamska_skala_i_historijske_primjene"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Logaritamke tablice, logaritamska skala i historijske primjene</span> </div> </a> <ul id="toc-Logaritamke_tablice,_logaritamska_skala_i_historijske_primjene-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analitička_svojstva" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analitička_svojstva"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Analitička svojstva</span> </div> </a> <button aria-controls="toc-Analitička_svojstva-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>Uključi/isključi podsekciju Analitička svojstva</span> </button> <ul id="toc-Analitička_svojstva-sublist" class="vector-toc-list"> <li id="toc-Logaritamska_funkcija" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logaritamska_funkcija"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Logaritamska funkcija</span> </div> </a> <ul id="toc-Logaritamska_funkcija-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Također_pogledajte" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Također_pogledajte"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Također pogledajte</span> </div> </a> <ul id="toc-Također_pogledajte-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bilješke" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bilješke"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bilješke</span> </div> </a> <ul id="toc-Bilješke-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Reference</span> </div> </a> <ul id="toc-Reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vanjski_linkovi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vanjski_linkovi"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Vanjski linkovi</span> </div> </a> <ul id="toc-Vanjski_linkovi-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="Sadržaj" 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="Uključi/isključi sadržaj" > <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">Uključi/isključi sadržaj</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">Logaritam</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="Idi na članak na drugom jeziku. Dostupno na 109 jezika" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-109" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">109 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Logaritme" title="Logaritme – afrikans" lang="af" hreflang="af" data-title="Logaritme" data-language-autonym="Afrikaans" data-language-local-name="afrikans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Logarithmus" title="Logarithmus – njemački (Švicarska)" lang="gsw" hreflang="gsw" data-title="Logarithmus" data-language-autonym="Alemannisch" data-language-local-name="njemački (Švicarska)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%8E%E1%8C%8B%E1%88%AA%E1%8B%9D%E1%88%9D" title="ሎጋሪዝም – amharski" lang="am" hreflang="am" data-title="ሎጋሪዝም" data-language-autonym="አማርኛ" data-language-local-name="amharski" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Logaritmo" title="Logaritmo – aragonski" lang="an" hreflang="an" data-title="Logaritmo" data-language-autonym="Aragonés" data-language-local-name="aragonski" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%84%D9%88%D8%BA%D8%A7%D8%B1%D9%8A%D8%AA%D9%85" title="لوغاريتم – arapski" lang="ar" hreflang="ar" data-title="لوغاريتم" data-language-autonym="العربية" data-language-local-name="arapski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%84%D9%88%DA%AD%D8%A7%D8%B1%D9%8A%D8%AA%D9%85" title="لوڭاريتم – Moroccan Arabic" lang="ary" hreflang="ary" data-title="لوڭاريتم" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%98%E0%A6%BE%E0%A6%A4%E0%A6%BE%E0%A6%82%E0%A6%95" title="ঘাতাংক – asamski" lang="as" hreflang="as" data-title="ঘাতাংক" data-language-autonym="অসমীয়া" data-language-local-name="asamski" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Logaritmu" title="Logaritmu – asturijski" lang="ast" hreflang="ast" data-title="Logaritmu" data-language-autonym="Asturianu" data-language-local-name="asturijski" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Loqarifm" title="Loqarifm – azerbejdžanski" lang="az" hreflang="az" data-title="Loqarifm" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – baškirski" lang="ba" hreflang="ba" data-title="Логарифм" data-language-autonym="Башҡортса" data-language-local-name="baškirski" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Luogar%C4%97tmos" title="Luogarėtmos – Samogitian" lang="sgs" hreflang="sgs" data-title="Luogarėtmos" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Central Bikol" lang="bcl" hreflang="bcl" data-title="Logaritmo" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D0%B0%D1%80%D1%8B%D1%84%D0%BC" title="Лагарыфм – bjeloruski" lang="be" hreflang="be" data-title="Лагарыфм" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%8F%D0%B3%D0%B0%D1%80%D1%8B%D1%82%D0%BC" title="Лягарытм – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лягарытм" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D1%8A%D0%BC" title="Логаритъм – bugarski" lang="bg" hreflang="bg" data-title="Логаритъм" data-language-autonym="Български" data-language-local-name="bugarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bjn mw-list-item"><a href="https://bjn.wikipedia.org/wiki/Logaritma" title="Logaritma – Banjar" lang="bjn" hreflang="bjn" data-title="Logaritma" data-language-autonym="Banjar" data-language-local-name="Banjar" class="interlanguage-link-target"><span>Banjar</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B2%E0%A6%97%E0%A6%BE%E0%A6%B0%E0%A6%BF%E0%A6%A6%E0%A6%AE" title="লগারিদম – bengalski" lang="bn" hreflang="bn" data-title="লগারিদম" data-language-autonym="বাংলা" data-language-local-name="bengalski" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Logaritm" title="Logaritm – bretonski" lang="br" hreflang="br" data-title="Logaritm" data-language-autonym="Brezhoneg" data-language-local-name="bretonski" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Логарифм" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Logaritme" title="Logaritme – katalonski" lang="ca" hreflang="ca" data-title="Logaritme" data-language-autonym="Català" data-language-local-name="katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%86%DA%AF%D8%A7%D8%B1%DB%8C%D8%AA%D9%85" title="لۆگاریتم – centralnokurdski" lang="ckb" hreflang="ckb" data-title="لۆگاریتم" data-language-autonym="کوردی" data-language-local-name="centralnokurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Logaritmus" title="Logaritmus – češki" lang="cs" hreflang="cs" data-title="Logaritmus" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – čuvaški" lang="cv" hreflang="cv" data-title="Логарифм" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Logarithm" title="Logarithm – velški" lang="cy" hreflang="cy" data-title="Logarithm" data-language-autonym="Cymraeg" data-language-local-name="velški" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Logaritme" title="Logaritme – danski" lang="da" hreflang="da" data-title="Logaritme" data-language-autonym="Dansk" data-language-local-name="danski" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Logarithmus" title="Logarithmus – njemački" lang="de" hreflang="de" data-title="Logarithmus" data-language-autonym="Deutsch" data-language-local-name="njemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Logaritma" title="Logaritma – Zazaki" lang="diq" hreflang="diq" data-title="Logaritma" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%AC%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82" title="Λογάριθμος – grčki" lang="el" hreflang="el" data-title="Λογάριθμος" data-language-autonym="Ελληνικά" data-language-local-name="grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Logar%C3%ACtem" title="Logarìtem – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Logarìtem" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437796 badge-featuredarticle mw-list-item" title="istaknuti članak"><a href="https://en.wikipedia.org/wiki/Logarithm" title="Logarithm – engleski" lang="en" hreflang="en" data-title="Logarithm" data-language-autonym="English" data-language-local-name="engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Logaritmo" title="Logaritmo – esperanto" lang="eo" hreflang="eo" data-title="Logaritmo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Logaritmo" title="Logaritmo – španski" lang="es" hreflang="es" data-title="Logaritmo" data-language-autonym="Español" data-language-local-name="španski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Logaritm" title="Logaritm – estonski" lang="et" hreflang="et" data-title="Logaritm" data-language-autonym="Eesti" data-language-local-name="estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Logaritmo" title="Logaritmo – baskijski" lang="eu" hreflang="eu" data-title="Logaritmo" data-language-autonym="Euskara" data-language-local-name="baskijski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Logaritmu" title="Logaritmu – Extremaduran" lang="ext" hreflang="ext" data-title="Logaritmu" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%84%DA%AF%D8%A7%D8%B1%DB%8C%D8%AA%D9%85" title="لگاریتم – perzijski" lang="fa" hreflang="fa" data-title="لگاریتم" data-language-autonym="فارسی" data-language-local-name="perzijski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Logaritmi" title="Logaritmi – finski" lang="fi" hreflang="fi" data-title="Logaritmi" data-language-autonym="Suomi" data-language-local-name="finski" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Logaritma" title="Logaritma – farski" lang="fo" hreflang="fo" data-title="Logaritma" data-language-autonym="Føroyskt" data-language-local-name="farski" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Logarithme" title="Logarithme – francuski" lang="fr" hreflang="fr" data-title="Logarithme" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Logartam" title="Logartam – irski" lang="ga" hreflang="ga" data-title="Logartam" data-language-autonym="Gaeilge" data-language-local-name="irski" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%B0%8D%E6%95%B8" title="對數 – Gan" lang="gan" hreflang="gan" data-title="對數" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Logaritm" title="Logaritm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Logaritm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – galicijski" lang="gl" hreflang="gl" data-title="Logaritmo" data-language-autonym="Galego" data-language-local-name="galicijski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9C%D7%95%D7%92%D7%A8%D7%99%D7%AA%D7%9D" title="לוגריתם – hebrejski" lang="he" hreflang="he" data-title="לוגריתם" data-language-autonym="עברית" data-language-local-name="hebrejski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B2%E0%A4%98%E0%A5%81%E0%A4%97%E0%A4%A3%E0%A4%95" title="लघुगणक – hindi" lang="hi" hreflang="hi" data-title="लघुगणक" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Logarithm" title="Logarithm – Fiji Hindi" lang="hif" hreflang="hif" data-title="Logarithm" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Logaritam" title="Logaritam – hrvatski" lang="hr" hreflang="hr" data-title="Logaritam" data-language-autonym="Hrvatski" data-language-local-name="hrvatski" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu badge-Q17437796 badge-featuredarticle mw-list-item" title="istaknuti članak"><a href="https://hu.wikipedia.org/wiki/Logaritmus" title="Logaritmus – mađarski" lang="hu" hreflang="hu" data-title="Logaritmus" data-language-autonym="Magyar" data-language-local-name="mađarski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BC%D5%B8%D5%A3%D5%A1%D6%80%D5%AB%D5%A9%D5%B4" title="Լոգարիթմ – armenski" lang="hy" hreflang="hy" data-title="Լոգարիթմ" data-language-autonym="Հայերեն" data-language-local-name="armenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Logarithmo" title="Logarithmo – interlingva" lang="ia" hreflang="ia" data-title="Logarithmo" data-language-autonym="Interlingua" data-language-local-name="interlingva" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logaritma" title="Logaritma – indonezijski" lang="id" hreflang="id" data-title="Logaritma" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Logaritmo" title="Logaritmo – ido" lang="io" hreflang="io" data-title="Logaritmo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Logri" title="Logri – islandski" lang="is" hreflang="is" data-title="Logri" data-language-autonym="Íslenska" data-language-local-name="islandski" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logaritmo" title="Logaritmo – italijanski" lang="it" hreflang="it" data-title="Logaritmo" data-language-autonym="Italiano" data-language-local-name="italijanski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AF%BE%E6%95%B0" title="対数 – japanski" lang="ja" hreflang="ja" data-title="対数" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Lagaridim" title="Lagaridim – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Lagaridim" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9A%E1%83%9D%E1%83%92%E1%83%90%E1%83%A0%E1%83%98%E1%83%97%E1%83%9B%E1%83%98" title="ლოგარითმი – gruzijski" lang="ka" hreflang="ka" data-title="ლოგარითმი" data-language-autonym="ქართული" data-language-local-name="gruzijski" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – kazaški" lang="kk" hreflang="kk" data-title="Логарифм" data-language-autonym="Қазақша" data-language-local-name="kazaški" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%9C%EA%B7%B8_(%EC%88%98%ED%95%99)" title="로그 (수학) – korejski" lang="ko" hreflang="ko" data-title="로그 (수학)" data-language-autonym="한국어" data-language-local-name="korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logarithmus" title="Logarithmus – latinski" lang="la" hreflang="la" data-title="Logarithmus" data-language-autonym="Latina" data-language-local-name="latinski" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Logaritmo" title="Logaritmo – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Logaritmo" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target"><span>Lingua Franca Nova</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Logaritm" title="Logaritm – lombardski" lang="lmo" hreflang="lmo" data-title="Logaritm" data-language-autonym="Lombard" data-language-local-name="lombardski" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Logaritmas" title="Logaritmas – litvanski" lang="lt" hreflang="lt" data-title="Logaritmas" data-language-autonym="Lietuvių" data-language-local-name="litvanski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Logaritms" title="Logaritms – latvijski" lang="lv" hreflang="lv" data-title="Logaritms" data-language-autonym="Latviešu" data-language-local-name="latvijski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Anisa" title="Anisa – malgaški" lang="mg" hreflang="mg" data-title="Anisa" data-language-autonym="Malagasy" data-language-local-name="malgaški" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="istaknuti članak"><a href="https://mk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Логаритам – makedonski" lang="mk" hreflang="mk" data-title="Логаритам" data-language-autonym="Македонски" data-language-local-name="makedonski" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B2%E0%B5%8B%E0%B4%97%E0%B4%B0%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ലോഗരിതം – malajalam" lang="ml" hreflang="ml" data-title="ലോഗരിതം" data-language-autonym="മലയാളം" data-language-local-name="malajalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B2%E0%A5%89%E0%A4%97%E0%A5%85%E0%A4%B0%E0%A4%BF%E0%A4%A6%E0%A4%AE" title="लॉगॅरिदम – marati" lang="mr" hreflang="mr" data-title="लॉगॅरिदम" data-language-autonym="मराठी" data-language-local-name="marati" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logaritma" title="Logaritma – malajski" lang="ms" hreflang="ms" data-title="Logaritma" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9C%E1%80%B1%E1%80%AC%E1%80%B7%E1%80%82%E1%80%9B%E1%80%85%E1%80%BA%E1%80%9E%E1%80%99%E1%80%BA" title="လော့ဂရစ်သမ် – burmanski" lang="my" hreflang="my" data-title="လော့ဂရစ်သမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burmanski" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Logarithmus" title="Logarithmus – donjonjemački" lang="nds" hreflang="nds" data-title="Logarithmus" data-language-autonym="Plattdüütsch" data-language-local-name="donjonjemački" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Logaritme" title="Logaritme – nizozemski" lang="nl" hreflang="nl" data-title="Logaritme" data-language-autonym="Nederlands" data-language-local-name="nizozemski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Logaritme" title="Logaritme – norveški (Nynorsk)" lang="nn" hreflang="nn" data-title="Logaritme" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Logaritme" title="Logaritme – norveški (Bokmal)" lang="nb" hreflang="nb" data-title="Logaritme" data-language-autonym="Norsk bokmål" data-language-local-name="norveški (Bokmal)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Logaritme" title="Logaritme – oksitanski" lang="oc" hreflang="oc" data-title="Logaritme" data-language-autonym="Occitan" data-language-local-name="oksitanski" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Loogarizimii" title="Loogarizimii – oromo" lang="om" hreflang="om" data-title="Loogarizimii" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A8%98%E0%A9%82%E0%A8%97%E0%A8%A3%E0%A8%95" title="ਲਘੂਗਣਕ – pandžapski" lang="pa" hreflang="pa" data-title="ਲਘੂਗਣਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pandžapski" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logarytm" title="Logarytm – poljski" lang="pl" hreflang="pl" data-title="Logarytm" data-language-autonym="Polski" data-language-local-name="poljski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%84%D8%A7%DA%AF%D8%B1%D8%AA%DA%BE%D9%85" title="لاگرتھم – Western Punjabi" lang="pnb" hreflang="pnb" data-title="لاگرتھم" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt badge-Q17437796 badge-featuredarticle mw-list-item" title="istaknuti članak"><a href="https://pt.wikipedia.org/wiki/Logaritmo" title="Logaritmo – portugalski" lang="pt" hreflang="pt" data-title="Logaritmo" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Logaritm" title="Logaritm – rumunski" lang="ro" hreflang="ro" data-title="Logaritm" data-language-autonym="Română" data-language-local-name="rumunski" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="istaknuti članak"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – ruski" lang="ru" hreflang="ru" data-title="Логарифм" data-language-autonym="Русский" data-language-local-name="ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – jakutski" lang="sah" hreflang="sah" data-title="Логарифм" data-language-autonym="Саха тыла" data-language-local-name="jakutski" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Logaritmu" title="Logaritmu – sicilijanski" lang="scn" hreflang="scn" data-title="Logaritmu" data-language-autonym="Sicilianu" data-language-local-name="sicilijanski" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh badge-Q70893996 mw-list-item" title=""><a href="https://sh.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Логаритам – srpskohrvatski" lang="sh" hreflang="sh" data-title="Логаритам" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpskohrvatski" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BD%E0%B6%9D%E0%B7%94_%E0%B6%9C%E0%B6%AB%E0%B6%9A" title="ලඝු ගණක – sinhaleški" lang="si" hreflang="si" data-title="ලඝු ගණක" data-language-autonym="සිංහල" data-language-local-name="sinhaleški" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Logarithm" title="Logarithm – Simple English" lang="en-simple" hreflang="en-simple" data-title="Logarithm" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Logaritmus" title="Logaritmus – slovački" lang="sk" hreflang="sk" data-title="Logaritmus" data-language-autonym="Slovenčina" data-language-local-name="slovački" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Logaritem" title="Logaritem – slovenski" lang="sl" hreflang="sl" data-title="Logaritem" data-language-autonym="Slovenščina" data-language-local-name="slovenski" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Daraunene" title="Daraunene – šona" lang="sn" hreflang="sn" data-title="Daraunene" data-language-autonym="ChiShona" data-language-local-name="šona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logaritmet" title="Logaritmet – albanski" lang="sq" hreflang="sq" data-title="Logaritmet" data-language-autonym="Shqip" data-language-local-name="albanski" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%82%D0%B0%D0%BC" title="Логаритам – srpski" lang="sr" hreflang="sr" data-title="Логаритам" data-language-autonym="Српски / srpski" data-language-local-name="srpski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Logaritm" title="Logaritm – švedski" lang="sv" hreflang="sv" data-title="Logaritm" data-language-autonym="Svenska" data-language-local-name="švedski" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Logi" title="Logi – svahili" lang="sw" hreflang="sw" data-title="Logi" data-language-autonym="Kiswahili" data-language-local-name="svahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AE%9F%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%88" title="மடக்கை – tamilski" lang="ta" hreflang="ta" data-title="மடக்கை" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%AD%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%B4%E0%B8%97%E0%B8%B6%E0%B8%A1" title="ลอการิทึม – tajlandski" lang="th" hreflang="th" data-title="ลอการิทึม" data-language-autonym="ไทย" data-language-local-name="tajlandski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Logaritmo" title="Logaritmo – tagalog" lang="tl" hreflang="tl" data-title="Logaritmo" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Logaritma" title="Logaritma – turski" lang="tr" hreflang="tr" data-title="Logaritma" data-language-autonym="Türkçe" data-language-local-name="turski" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – tatarski" lang="tt" hreflang="tt" data-title="Логарифм" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarski" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – ukrajinski" lang="uk" hreflang="uk" data-title="Логарифм" data-language-autonym="Українська" data-language-local-name="ukrajinski" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%84%D8%A7%DA%AF%D8%B1%D8%AA%DA%BE%D9%85" title="لاگرتھم – urdu" lang="ur" hreflang="ur" data-title="لاگرتھم" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Logarifm" title="Logarifm – uzbečki" lang="uz" hreflang="uz" data-title="Logarifm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="istaknuti članak"><a href="https://vi.wikipedia.org/wiki/Logarit" title="Logarit – vijetnamski" lang="vi" hreflang="vi" data-title="Logarit" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Logaritmo" title="Logaritmo – varej" lang="war" hreflang="war" data-title="Logaritmo" data-language-autonym="Winaray" data-language-local-name="varej" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%AF%B9%E6%95%B0" title="对数 – Wu kineski" lang="wuu" hreflang="wuu" data-title="对数" data-language-autonym="吴语" data-language-local-name="Wu kineski" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%9C%D7%90%D7%92%D7%90%D7%A8%D7%99%D7%98%D7%9D" title="לאגאריטם – jidiš" lang="yi" hreflang="yi" data-title="לאגאריטם" data-language-autonym="ייִדיש" data-language-local-name="jidiš" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%AF%B9%E6%95%B0" title="对数 – kineski" lang="zh" hreflang="zh" data-title="对数" data-language-autonym="中文" data-language-local-name="kineski" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/T%C3%B9i-s%C3%B2%CD%98" title="Tùi-sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Tùi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%B0%8D%E6%95%B8" title="對數 – kantonski" lang="yue" hreflang="yue" data-title="對數" data-language-autonym="粵語" data-language-local-name="kantonski" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11197#sitelinks-wikipedia" title="Uredi međujezičke linkove" class="wbc-editpage">Uredi veze</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Imenski prostori"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Logaritam" title="Prikaži stranicu sa 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title="Spisak svih članaka koji su povezani s ovim [j]" accesskey="j"><span>Šta vodi ovamo</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Posebno:Srodne_izmjene/Logaritam" rel="nofollow" title="Nedavne izmjene na stranicama koje su povezane sa ovom [k]" accesskey="k"><span>Srodne izmjene</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:Upload" title="Postavi slike i druge medije [u]" accesskey="u"><span>Postavi datoteku</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Posebno:Posebne_stranice" title="Spisak svih posebnih stranica [q]" accesskey="q"><span>Posebne stranice</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Logaritam&oldid=3608070" title="Trajni link ove verzije stranice"><span>Trajni link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Logaritam&action=info" title="Više informacija o ovoj stranici"><span>Informacije 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navigation-not-searchable">Razlikovati: <a href="/wiki/Algoritam" title="Algoritam">Algoritam</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Binary_logarithm_plot_with_ticks.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/300px-Binary_logarithm_plot_with_ticks.svg.png" decoding="async" width="300" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/450px-Binary_logarithm_plot_with_ticks.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Binary_logarithm_plot_with_ticks.svg/600px-Binary_logarithm_plot_with_ticks.svg.png 2x" data-file-width="408" data-file-height="325" /></a><figcaption><a href="/wiki/Grafik_funkcije" title="Grafik funkcije">Grafik</a> logaritma za bazu 2 presijeca <a href="/w/index.php?title=X-osa&action=edit&redlink=1" class="new" title="X-osa (stranica ne postoji)"><i>x</i>-osu</a> (horizontalna osa) u tački 1 i prolazi dalje kroz tačke sa <a href="/w/index.php?title=Koordinata&action=edit&redlink=1" class="new" title="Koordinata (stranica ne postoji)">koordinatama</a> <span class="nowrap">(2, 1)</span>, <span class="nowrap">(4, 2)</span>, i <span class="nowrap">(8, 3)</span>. Naprimjer, <span class="nowrap">log<sub>2</sub>(8) = 3</span>, jer je <span class="nowrap">2<sup>3</sup> = 8.</span> Kriva se neprestano približava <i>y</i>-osi, ali <a href="/w/index.php?title=Asimptota&action=edit&redlink=1" class="new" title="Asimptota (stranica ne postoji)">je nikad ne dodiruje niti presijeca</a>.</figcaption></figure> <p>U <a href="/wiki/Matematika" title="Matematika">matematici</a>, <b>logaritam</b> je <a href="/w/index.php?title=Inverzna_operacija&action=edit&redlink=1" class="new" title="Inverzna operacija (stranica ne postoji)">inverzna operacija</a> za <a href="/w/index.php?title=Eksponencija&action=edit&redlink=1" class="new" title="Eksponencija (stranica ne postoji)">eksponenciju</a>. To znači da je logaritam broja ustvari <a href="/wiki/Eksponent" title="Eksponent">eksponent</a> za koju određena fiksna vrijednost, <a href="/w/index.php?title=Baza_(eksponencija)&action=edit&redlink=1" class="new" title="Baza (eksponencija) (stranica ne postoji)">baza</a>, mora biti stepenovana da proizvede taj broj. U jednostavnim slučajevima logaritam se smatra ponovljenim množenjem. Naprimjer, baza <span class="texhtml">10</span> logaritma od <span class="texhtml">1000</span> jeste <span class="texhtml">3</span>, jer <span class="texhtml">10</span> stepenovan brojem <span class="texhtml">3</span> daje <span class="texhtml">1000</span> (<span class="texhtml">1000 = 10 × 10 × 10 = 10<sup>3</sup></span>); multiplikacija se ponavlja tri puta. Općenitije, eksponencija dopušta bilo kojem <a href="/wiki/Realni_broj" class="mw-redirect" title="Realni broj">realnom broju</a> da se podigne na bilo koji realan stepen, uvijek proizvodeći pozitivan rezultat, tako da se logaritam može izračunati za bilo koja dva pozitivna realna broja <span class="texhtml"><i>b</i></span> i <span class="texhtml"><i>x</i></span> gdje <span class="texhtml"><i>b</i></span> nije jednako <span class="texhtml">1</span>. Logaritam od <span class="texhtml"><i>x</i></span> za <i>bazu</i> <span class="texhtml"><i>b</i></span>, piše se <span class="texhtml">log<sub><i>b</i></sub>(<i>x</i>)</span>, jedinstven je realan broj <span class="texhtml"><i>y</i></span> takav da vrijedi </p> <dl><dd><span class="texhtml"><i>b</i><sup><i>y</i></sup> = <i>x</i></span>.</dd></dl> <p>Naprimjer, jer je <span class="texhtml">64 = 2<sup><sup>6</sup></sup></span>, onda je: </p> <dl><dd><span class="texhtml">log<sub>2</sub>(64) = 6</span></dd></dl> <p>Logaritam za bazu <span class="texhtml">10</span> (gdje je <span class="texhtml"><i>b</i> = 10</span>) zove se <a href="/w/index.php?title=Op%C4%87i_algoritam&action=edit&redlink=1" class="new" title="Opći algoritam (stranica ne postoji)">opći algoritam</a> i ima nekoliko primjena u nauci i inženjerstvu. <a href="/wiki/Prirodni_logaritam" title="Prirodni logaritam">Prirodni logaritam</a> ima <a href="/w/index.php?title=E_(matemati%C4%8Dka_konstanta)&action=edit&redlink=1" class="new" title="E (matematička konstanta) (stranica ne postoji)">broj <span style="white-space:nowrap"><span class="texhtml"><i>e</i></span></span></a> (<span class="texhtml">≈ 2.718</span>) kao bazu; njegova primjena je raširena u matematici i <a href="/wiki/Fizika" title="Fizika">fizici</a>, zbog svog jednostavnijeg <a href="/wiki/Izvod" title="Izvod">izvoda</a>. <a href="/w/index.php?title=Binarni_logaritam&action=edit&redlink=1" class="new" title="Binarni logaritam (stranica ne postoji)">Binarni logaritam</a> koristi bazu <span class="texhtml">2</span> (gdje je <span class="texhtml"><i>b</i> = 2</span>) i često se koristi u <a href="/wiki/Ra%C4%8Dunarstvo" title="Računarstvo">računarstvu</a>. </p><p>Logaritme je uveo <a href="/wiki/John_Napier" title="John Napier">John Napier</a> početkom 17. vijeka radi pojednostavljenja proračuna. Oni se uveliko koriste od strane navigatora, naučnika, inženjera i ostalih kako bi se računarski proračuni izvršavali mnogo lakše, koristeći <a href="/w/index.php?title=Logaritmar&action=edit&redlink=1" class="new" title="Logaritmar (stranica ne postoji)">logaritmar</a> i <a href="/w/index.php?title=Matemati%C4%8Dka_tabla&action=edit&redlink=1" class="new" title="Matematička tabla (stranica ne postoji)">logaritamske tablice</a>. Zamorno višecifreno množenje mogu zamijeniti tablice s jednostavnim sabiranjem zbog činjenice — veoma važne — da je logaritamski <a href="/w/index.php?title=Proizvod_(matematika)&action=edit&redlink=1" class="new" title="Proizvod (matematika) (stranica ne postoji)">proizvod</a> zapravo <a href="/w/index.php?title=Zbir&action=edit&redlink=1" class="new" title="Zbir (stranica ne postoji)">zbir</a> logaritama faktora: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y),\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd4d4589711179f7c3c343857466028d67ff7cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.099ex; height:2.843ex;" alt="{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y),\,}"></span></dd></dl> <p>gdje su <span class="texhtml"><i>b</i></span>, <span class="texhtml"><i>x</i></span> i <span class="texhtml"><i>y</i></span> svi pozitivni i <span class="texhtml"><i>b</i> ≠ 1</span>. Današnji pojam logaritma dolazi od <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonharda Eulera</a>, koji je napravio vezu između logaritama i <a href="/wiki/Eksponencijalna_funkcija" title="Eksponencijalna funkcija">eksponencijalne funkcije</a> u 18. vijeku. </p><p><a href="/w/index.php?title=Logaritamska_skala&action=edit&redlink=1" class="new" title="Logaritamska skala (stranica ne postoji)">Logaritamska skala</a> smanjuje širok spektar veličina na manje prostora. Naprimjer, <a href="/w/index.php?title=Decibel&action=edit&redlink=1" class="new" title="Decibel (stranica ne postoji)">decibel</a> je <a href="/wiki/Mjerna_jedinica" title="Mjerna jedinica">mjerna jedinica</a> jačine signala snage log-odnosa i amplitude log-odnosa (od kojih je <a href="/w/index.php?title=Zvu%C4%8Dni_pritisak&action=edit&redlink=1" class="new" title="Zvučni pritisak (stranica ne postoji)">zvučni pritisak</a> čest primjer). U hemiji, <a href="/wiki/PH" title="PH">pH</a> je logaritamska mjera za <a href="/wiki/Kiselina" class="mw-redirect" title="Kiselina">kiselost</a> <a href="/w/index.php?title=Vodena_otopina&action=edit&redlink=1" class="new" title="Vodena otopina (stranica ne postoji)">vodene otopine</a>. Logaritmi su uobičajena u naučnim <a href="/w/index.php?title=Formula&action=edit&redlink=1" class="new" title="Formula (stranica ne postoji)">formulama</a>, te u mjerama <a href="/w/index.php?title=Ra%C4%8Dunarska_teorija_slo%C5%BEenosti&action=edit&redlink=1" class="new" title="Računarska teorija složenosti (stranica ne postoji)">kompleksnosti algoritama</a> i geometrijskih objekata zvanih <a href="/wiki/Fraktal" title="Fraktal">fraktali</a>. Oni opisuju <a href="/w/index.php?title=Interval_(muzika)&action=edit&redlink=1" class="new" title="Interval (muzika) (stranica ne postoji)">muzičke intervale</a>, pojavljuju se u formulama brojeći <a href="/w/index.php?title=Prosti_broj&action=edit&redlink=1" class="new" title="Prosti broj (stranica ne postoji)">proste brojeve</a>, informišu neke modele u <a href="/w/index.php?title=Psihofizika&action=edit&redlink=1" class="new" title="Psihofizika (stranica ne postoji)">psihofizici</a>, te mogu pomoći u <a href="/w/index.php?title=Forenzi%C4%8Dko_ra%C4%8Dunovodstvo&action=edit&redlink=1" class="new" title="Forenzičko računovodstvo (stranica ne postoji)">forenzičkom računovodstvu</a>. </p><p>Na isti način kako logaritam služi <a href="/w/index.php?title=Eksponencija&action=edit&redlink=1" class="new" title="Eksponencija (stranica ne postoji)">eksponenciji</a>, <a href="/w/index.php?title=Kompleksni_logaritam&action=edit&redlink=1" class="new" title="Kompleksni logaritam (stranica ne postoji)">kompleksni logaritam</a> je <a href="/wiki/Inverzna_funkcija" title="Inverzna funkcija">inverzna funkcija</a> eksponencijalne funkcije primijenjene na <a href="/wiki/Kompleksni_broj" class="mw-redirect" title="Kompleksni broj">kompleksne brojeve</a>. <a href="/w/index.php?title=Diskretni_logaritam&action=edit&redlink=1" class="new" title="Diskretni logaritam (stranica ne postoji)">Diskretni logaritam</a> je naredna varijanta; koristi se u <a href="/w/index.php?title=Asimetri%C4%8Dna_kriptografija&action=edit&redlink=1" class="new" title="Asimetrična kriptografija (stranica ne postoji)">asimetričnoj kriptografiji</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Motivacija_i_definicija">Motivacija i definicija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=1" title="Uredi odlomak "Motivacija i definicija"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=1" title="Edit section's source code: Motivacija i definicija"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ideja logaritama je da obrnu operaciju <a href="/w/index.php?title=Eksponencija&action=edit&redlink=1" class="new" title="Eksponencija (stranica ne postoji)">eksponencije</a>, to jeste, stepenovanje broja određenim stepenom. Naprimjer, treći stepen (ili <a href="/w/index.php?title=Kocka_(algebra)&action=edit&redlink=1" class="new" title="Kocka (algebra) (stranica ne postoji)">kocka</a>) od 2 jeste 8, jer je 8 proizvod tri faktora 2: </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 2^{3}=2\times 2\times 2=8.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>=</mo> <mn>8.</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}=2\times 2\times 2=8.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a7999b3f20651d1bdb4968820651a3137dd7e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.778ex; height:2.676ex;" alt="{\displaystyle 2^{3}=2\times 2\times 2=8.\,}"></span></dd></dl> <p>To znači daje logaritam od 8 sa bazom 2 upravo 3, tako da je log<sub>2</sub> 8 = 3. </p> <div class="mw-heading mw-heading3"><h3 id="Eksponencija">Eksponencija</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=2" title="Uredi odlomak "Eksponencija"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=2" title="Edit section's source code: Eksponencija"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Treći stepen nekog broja <i>b</i> jeste proizvod tri faktora od <i>b</i>. Općenitije, stepenovanjem <i>b</i> na <span class="nowrap"><i>n</i>-ti</span> stepen, gdje je <i>n</i> <a href="/wiki/Prirodni_broj" class="mw-redirect" title="Prirodni broj">prirodni broj</a>, radi se množenjem <i>n</i> faktora od <i>b</i>. <span class="nowrap"><i>n</i>-ti</span> stepen od <i>b</i> se piše kao <i>b</i><sup><i>n</i></sup>, tako da je </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}=\underbrace {b\times b\times \cdots \times b} _{n{\text{ factors}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>b</mi> <mo>×</mo> <mi>b</mi> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mi>b</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> factors</mtext> </mrow> </mrow> </munder> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{n}=\underbrace {b\times b\times \cdots \times b} _{n{\text{ factors}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ffd1659b7f9831ec182fb01ab6e50aabbdaabee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:20.198ex; height:5.843ex;" alt="{\displaystyle b^{n}=\underbrace {b\times b\times \cdots \times b} _{n{\text{ factors}}}.}"></span></dd></dl> <p>Eksponencija se može proširiti na <i>b</i><sup><i>y</i></sup>, gdje je <i>b</i> pozitivni broj i <i>eksponent</i> <i>y</i> je bilo koji <a href="/wiki/Realni_broj" class="mw-redirect" title="Realni broj">realni broj</a>. Naprimjer, <i>b</i><sup>−1</sup> je recipročan od <i>b</i>, to jeste, <span class="nowrap">1/<i>b</i></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Definicija">Definicija</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=3" title="Uredi odlomak "Definicija"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=3" title="Edit section's source code: Definicija"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Logaritam</i> pozitivnog realnog broja <i>x</i> sa bazom <i>b</i>, pozitivni realan broj nejednak sa 1<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup>, jeste eksponent kojim <i>b</i> mora biti stepenovan da se dobije <i>x</i>. Drugim riječima, logaritam od <i>x</i> za bazu <i>b</i> je rješenje <i>y</i> za jednačinu<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{y}=x.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mi>x</mi> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{y}=x.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b10f7131bb0093b393af99e97cf899cea963ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.509ex; height:2.343ex;" alt="{\displaystyle b^{y}=x.\,}"></span></dd></dl> <p>Logaritam je opisan "log<sub><i>b</i></sub>(<i>x</i>)" (čita se "logaritam od <i>x</i> za bazu <i>b</i>". U jednačini <i>y</i> = log<sub><i>b</i></sub>(<i>x</i>), vrijednost <i>y</i> je odgovor na pitanje "Na koji stepen mora biti <i>b</i> dignut, da bi se dobio <i>x</i>?". Ovo pitanje može također biti upućeno (sa bogatijim odgovorom) za <a href="/wiki/Kompleksni_broj" class="mw-redirect" title="Kompleksni broj">kompleksne brojeve</a>, što je pokazano u sekciji <a href="#Kompleksni_logaritam">"Kompleksni logaritam"</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Primjeri">Primjeri</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=4" title="Uredi odlomak "Primjeri"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=4" title="Edit section's source code: Primjeri"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Naprimjer, <span class="nowrap">log<sub>2</sub>(16) = 4</span>, pošto je <span class="nowrap">2<sup>4</sup> = 2 ×2 × 2 × 2</span> = 16. Logaritmi također mogu biti negativni: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}\!\left({\frac {1}{2}}\right)=-1,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}\!\left({\frac {1}{2}}\right)=-1,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76802a3a67e36967a79144f0ac096d5b73b712c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.549ex; height:6.176ex;" alt="{\displaystyle \log _{2}\!\left({\frac {1}{2}}\right)=-1,\,}"></span></dd></dl> <p>pošto je </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 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2654a9bb9118970cbc9ad5a2e5d46d822609d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.39ex; height:5.676ex;" alt="{\displaystyle 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}"></span></dd></dl> <p>Treći primjer: log<sub>10</sub>(150) je približno 2.176, što leži između 2 i 3, kao što 150 leži između <span class="nowrap">10<sup>2</sup> = 100</span> i <span class="nowrap">10<sup>3</sup> = 1000</span>. Konačno, za bilo koju bazu <i>b</i>, <span class="nowrap">log<sub><i>b</i></sub>(<i>b</i>) = 1</span> i <span class="nowrap">log<sub><i>b</i></sub>(1) = 0</span>, pošto važi <span class="nowrap"><i>b</i><sup>1</sup> = <i>b</i></span> i <span class="nowrap"><i>b</i><sup>0</sup> = 1</span>, redom. </p> <div class="mw-heading mw-heading2"><h2 id="Logaritamski_identiteti">Logaritamski identiteti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=5" title="Uredi odlomak "Logaritamski identiteti"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=5" title="Edit section's source code: Logaritamski identiteti"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Glavni članak: <a href="/w/index.php?title=Spisak_logaritamskih_identiteta&action=edit&redlink=1" class="new" title="Spisak logaritamskih identiteta (stranica ne postoji)">Spisak logaritamskih identiteta</a></div> <p>Nekoliko važnih formula, ponekad zvanih <i>logaritamski identiteti</i> ili <i>logaritamski zakoni</i>, vežu logaritme međusobno.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Proizvod,_koeficijent,_stepen_i_korijen"><span id="Proizvod.2C_koeficijent.2C_stepen_i_korijen"></span>Proizvod, koeficijent, stepen i korijen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=6" title="Uredi odlomak "Proizvod, koeficijent, stepen i korijen"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=6" title="Edit section's source code: Proizvod, koeficijent, stepen i korijen"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Logaritam proizvoda jeste suma logaritama brojeva koji se množe; logaritam odnosa dva broja jednaka je razlici logaritama. Logaritam od <span class="nowrap"><i>p</i>-tog</span> stepena broja je <i>p</i> puta logaritam samog broja; logaritam od <span class="nowrap"><i>p</i>-tog</span> korijena je logaritam broja podijeljen sa <i>p</i>. Slijedeća tabela pokazuje ove identitete sa primjerima. Svaki od ovih identiteta može biti izveden nakon smjene definicije logaritma <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=b^{\log _{b}(x)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=b^{\log _{b}(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d465f89dbf4ec4056a55c45b09c3de7922a6cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.716ex; height:2.843ex;" alt="{\displaystyle x=b^{\log _{b}(x)}}"></span> ili <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=b^{\log _{b}(y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=b^{\log _{b}(y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba73f54b277e9f747e825f3a8f4f4b0272d3ffd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.418ex; height:3.176ex;" alt="{\displaystyle y=b^{\log _{b}(y)}}"></span> na lijevoj strani. </p> <table class="wikitable" style="margin: 0 auto;"> <tbody><tr> <th></th> <th>Formula</th> <th>Primjer </th></tr> <tr> <td>proizvod</td> <td><cite id="labegarithmProducts"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a72b4b7ba4c487ba5c15587d2eff610355605901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.065ex; height:2.843ex;" alt="{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y)}"></span></cite></td> <td><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 _{3}(243)=\log _{3}(9\cdot 27)=\log _{3}(9)+\log _{3}(27)=2+3=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>243</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>9</mn> <mo>⋅</mo> <mn>27</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>27</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{3}(243)=\log _{3}(9\cdot 27)=\log _{3}(9)+\log _{3}(27)=2+3=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e436f1647d5647a0bd981927708de198a516f4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.044ex; height:2.843ex;" alt="{\displaystyle \log _{3}(243)=\log _{3}(9\cdot 27)=\log _{3}(9)+\log _{3}(27)=2+3=5}"></span> </td></tr> <tr> <td>koeficijent</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}\!\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}\!\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7981f7ecdd8c824c9b6cad205a1ecd73e47e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.358ex; height:6.176ex;" alt="{\displaystyle \log _{b}\!\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y)}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}(16)=\log _{2}\!\left({\frac {64}{4}}\right)=\log _{2}(64)-\log _{2}(4)=6-2=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>16</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>64</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>64</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}(16)=\log _{2}\!\left({\frac {64}{4}}\right)=\log _{2}(64)-\log _{2}(4)=6-2=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fdf7ca5b5a795e513e20979b01fed8090976580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.488ex; height:6.176ex;" alt="{\displaystyle \log _{2}(16)=\log _{2}\!\left({\frac {64}{4}}\right)=\log _{2}(64)-\log _{2}(4)=6-2=4}"></span> </td></tr> <tr> <td>stepen</td> <td><cite id="labelLogarithmPowers"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(x^{p})=p\log _{b}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x^{p})=p\log _{b}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e081c1847c72f5c52c0361230fc7ecf8db17986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.811ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x^{p})=p\log _{b}(x)}"></span></cite></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}(64)=\log _{2}(2^{6})=6\log _{2}(2)=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>64</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}(64)=\log _{2}(2^{6})=6\log _{2}(2)=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103d4ca851421e48d662ebbb8733be07c52b43c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.217ex; height:3.176ex;" alt="{\displaystyle \log _{2}(64)=\log _{2}(2^{6})=6\log _{2}(2)=6}"></span> </td></tr> <tr> <td>korijen</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}(x)}{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}(x)}{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2499a9e2a1b1e9e77ccb8dda608b297e0b943f19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.545ex; height:6.176ex;" alt="{\displaystyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}(x)}{p}}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1000</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>1000</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdaede671aaadfc7608d97f962e252497b46c280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.357ex; height:5.176ex;" alt="{\displaystyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Izmjena_baze">Izmjena baze</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=7" title="Uredi odlomak "Izmjena baze"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=7" title="Edit section's source code: Izmjena baze"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Logaritam log<sub><i>b</i></sub>(<i>x</i>) može biti izračunat iz logaritama od <i>x</i> i <i>b</i> uzimajući u obzir proizvoljnu bazu <i>k</i> preko sljedeće formule: </p> <dl><dd><cite id="labelLogarithmBaseChange"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5b646b75b8d314ff3657731a96904ccb488d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.216ex; height:6.509ex;" alt="{\displaystyle \log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}.\,}"></span></cite></dd></dl> <p>Obični <a href="/w/index.php?title=Nau%C4%8Dni_digitron&action=edit&redlink=1" class="new" title="Naučni digitron (stranica ne postoji)">naučni digitron</a> računa logaritme sa bazama 10 i <a href="/w/index.php?title=E_(matemati%C4%8Dka_konstanta)&action=edit&redlink=1" class="new" title="E (matematička konstanta) (stranica ne postoji)">konstantom <i>e</i></a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Logaritmi s obzirom na bilo koju bazu <i>b</i> mogu se odrediti korištenjem bilo kojih od dva logaritama preko prethodne formule: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}={\frac {\log _{e}(x)}{\log _{e}(b)}}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}={\frac {\log _{e}(x)}{\log _{e}(b)}}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9890ffb1f927906d91affa2a12429e7366f473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.048ex; height:6.509ex;" alt="{\displaystyle \log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}={\frac {\log _{e}(x)}{\log _{e}(b)}}.\,}"></span></dd></dl> <p>Neka je dat broj <i>x</i> i njegov logaritam log<sub><i>b</i></sub>(<i>x</i>) za nepoznatu bazu <i>b</i>, baza je data sa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=x^{\frac {1}{\log _{b}(x)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=x^{\frac {1}{\log _{b}(x)}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ee0a60595a0bff6dbd9e74808033592ce0d24a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.355ex; height:4.176ex;" alt="{\displaystyle b=x^{\frac {1}{\log _{b}(x)}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Određene_baze"><span id="Odre.C4.91ene_baze"></span>Određene baze</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=8" title="Uredi odlomak "Određene baze"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=8" title="Edit section's source code: Određene baze"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Među svim izborima za bazu, tri su posebno česta. To su <i>b</i> = 10, <i>b</i> = <a href="/w/index.php?title=E_(matemati%C4%8Dka_konstanta)&action=edit&redlink=1" class="new" title="E (matematička konstanta) (stranica ne postoji)"><i>e</i></a> (<a href="/wiki/Iracionalan_broj" title="Iracionalan broj">iracionalna</a> matematička konstanta ≈ 2,71828), i <i>b</i> = 2. U <a href="/w/index.php?title=Maatemati%C4%8Dka_analiza&action=edit&redlink=1" class="new" title="Maatematička analiza (stranica ne postoji)">matematičkoj analizi</a>, logaritam za bazu <i>e</i> je raširen zbog svojih određenih analitičkih svojstava objašnjenih ispod. U drugu ruku, algoritmi s bazom 10 su jednostavni za koristiti za ručne proračune u decimalnom brojnom sistemu:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}(10x)=\log _{10}(10)+\log _{10}(x)=1+\log _{10}(x).\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <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> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(10x)=\log _{10}(10)+\log _{10}(x)=1+\log _{10}(x).\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09814dd4764899e69a92e302a8987c58726d8f82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.923ex; height:2.843ex;" alt="{\displaystyle \log _{10}(10x)=\log _{10}(10)+\log _{10}(x)=1+\log _{10}(x).\ }"></span></dd></dl> <p>Tako, log<sub>10</sub>(<i>x</i>) je vezan za broj <a href="/w/index.php?title=Decimalni_broj&action=edit&redlink=1" class="new" title="Decimalni broj (stranica ne postoji)">decimalnih brojeva</a> pozitivnog cijelog broja <i>x</i>: broj brojki je najmanji <a href="/wiki/Cijeli_broj" title="Cijeli broj">cijeli broj</a> striktno veći od log<sub>10</sub>(<i>x</i>).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Naprimjer, log<sub>10</sub>(1430) je približno 3,15. Slijedeći cijeli broj je 4, što je broj brojki od 1430. I prirodni logaritam i logaritam za bazu 2 se koriste u <a href="/w/index.php?title=Informaciona_teorija&action=edit&redlink=1" class="new" title="Informaciona teorija (stranica ne postoji)">informacionoj teoriji</a>, što odgovara upotrebi <a href="/w/index.php?title=Nat_(jedinica)&action=edit&redlink=1" class="new" title="Nat (jedinica) (stranica ne postoji)">natu</a> ili <a href="/wiki/Bit" title="Bit">bitovima</a> kao osnovnim jedinicama informacije, respektivno.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Binarni logaritmi su također korišteni u <a href="/wiki/Ra%C4%8Dunarstvo" title="Računarstvo">računarstvu</a>, gdje je <a href="/w/index.php?title=Binarni_brojni_sistem&action=edit&redlink=1" class="new" title="Binarni brojni sistem (stranica ne postoji)">binarni brojni sistem</a> sveprisutan, u <a href="/w/index.php?title=Muzi%C4%8Dka_teorija&action=edit&redlink=1" class="new" title="Muzička teorija (stranica ne postoji)">muzičkoj teoriji</a>, gdje je omjer visine tona dva (<a href="/wiki/Oktava" title="Oktava">oktava</a>) sveprisutan i <a href="/w/index.php?title=Cent_(muzika)&action=edit&redlink=1" class="new" title="Cent (muzika) (stranica ne postoji)">cent</a> je binarni logaritam (umanjen za 1200) od odnosa između dva susjedna jednako smirena tona, te u <a href="/wiki/Fotografija" title="Fotografija">fotografiji</a> za mjerenje vrijednosti izlaganja.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Slijedeća tabela pokazuje česte notacije za logaritme za ove baze i polja gdje se koriste. Dosta disciplina piše log(<i>x</i>) umjesto log<sub><i>b</i></sub>(<i>x</i>), kada se izabrana baza može odrediti iz konteksta. Notacija <sup><i>b</i></sup>log(<i>x</i>) također se pojavljuje.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Kolona "ISO notacija" pokazuje preporuke od <a href="/w/index.php?title=Me%C4%91unarodna_organizacija_za_strandardizaciju&action=edit&redlink=1" class="new" title="Međunarodna organizacija za strandardizaciju (stranica ne postoji)">ISO organizacije</a>, (<a href="/w/index.php?title=ISO_31-11&action=edit&redlink=1" class="new" title="ISO 31-11 (stranica ne postoji)">ISO 31-11</a>).<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" style="text-align:center; margin:1em auto 1em auto;"> <tbody><tr> <th scope="col">Baza <i>b</i> </th> <th scope="col">Ime za log<sub><i>b</i></sub>(<i>x</i>) </th> <th scope="col">ISO notacija </th> <th scope="col">Druge notacije </th> <th scope="col">Koristi se u </th></tr> <tr> <th scope="row">2 </th> <td><a href="/w/index.php?title=Binarni_logaritam&action=edit&redlink=1" class="new" title="Binarni logaritam (stranica ne postoji)">binarni logaritam</a> </td> <td>lb(<i>x</i>)<sup id="cite_ref-gullberg_11-0" class="reference"><a href="#cite_note-gullberg-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td> <td>ld(<i>x</i>), log(<i>x</i>), lg(<i>x</i>),<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> log2(<i>x</i>) </td> <td><a href="/wiki/Ra%C4%8Dunarstvo" title="Računarstvo">računarstvo</a>, <a href="/w/index.php?title=Informaciona_teorija&action=edit&redlink=1" class="new" title="Informaciona teorija (stranica ne postoji)">informaciona teorija</a>, <a href="/w/index.php?title=Muzi%C4%8Dka_teorija&action=edit&redlink=1" class="new" title="Muzička teorija (stranica ne postoji)">muzička teorija</a>, <a href="/wiki/Fotografija" title="Fotografija">fotografija</a> </td></tr> <tr> <th scope="row"><i>e</i> </th> <td><a href="/wiki/Prirodni_logaritam" title="Prirodni logaritam">prirodni logaritam</a> </td> <td>ln(<i>x</i>)<sup id="cite_ref-adaa_16-0" class="reference"><a href="#cite_note-adaa-16"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </td> <td>log(<i>x</i>)<br />(u matematici <sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> i više <a href="/wiki/Programski_jezik" title="Programski jezik">programskih jezika</a><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>nb 3<span class="cite-bracket">]</span></a></sup>) </td> <td>matematika, fizika, hemija,<br /><a href="/wiki/Statistika" title="Statistika">statistika</a>, <a href="/wiki/Ekonomija" title="Ekonomija">ekonomija</a>, informaciona teorija, i neka polja inženjerstva </td></tr> <tr> <th scope="row">10 </th> <td><a href="/w/index.php?title=Op%C4%87i_logaritam&action=edit&redlink=1" class="new" title="Opći logaritam (stranica ne postoji)">opći logaritam</a> </td> <td>lg(<i>x</i>) </td> <td>log(<i>x</i>), log<sub>10</sub>(<i>x</i>)<br />(u inženjerstvu, biologiji, astronomiji) </td> <td>različita <a href="/wiki/In%C5%BEenjerstvo" title="Inženjerstvo">inženjerska polja</a> (pogledati <a href="/w/index.php?title=Decibel&action=edit&redlink=1" class="new" title="Decibel (stranica ne postoji)">decibel</a> i ostalo ispod), <br />logaritamske <a href="/w/index.php?title=Matemati%C4%8Dka_tablica&action=edit&redlink=1" class="new" title="Matematička tablica (stranica ne postoji)">tablice</a>, ručni <a href="/w/index.php?title=Digitron&action=edit&redlink=1" class="new" title="Digitron (stranica ne postoji)">digitron</a>, <a href="/wiki/Spektroskopija" title="Spektroskopija">spektroskopija</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Historija">Historija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=9" title="Uredi odlomak "Historija"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=9" title="Edit section's source code: Historija"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Glavni članak: <a href="/w/index.php?title=Hitorija_logaritama&action=edit&redlink=1" class="new" title="Hitorija logaritama (stranica ne postoji)">Hitorija logaritama</a></div> <p><b>Historija logaritama</b> u Evropi u 17. vijeku jeste otkriće nove <a href="/wiki/Funkcija_(matematika)" title="Funkcija (matematika)">funkcije</a> koja je proširila stvarnost analize iza opsega algebarske metode. Metodu logaritama je javno objavio <a href="/wiki/John_Napier" title="John Napier">John Napier</a> 1614. godine, u knjizi naslova <i>Mirifici Logarithmorum Canonis Descriptio</i> (<i>Opis čudesnog pravila logaritama</i>).<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Prije Napierovog izuma, postojale su slične tehnike sličnog opsega, kao što su prostafereza ili korištenje tablica progresije, koje je ekstenzivno razvio <a href="/w/index.php?title=Jost_B%C3%BCrgi&action=edit&redlink=1" class="new" title="Jost Bürgi (stranica ne postoji)">Jost Bürgi</a> oko 1600. godine.<sup id="cite_ref-folkerts_21-0" class="reference"><a href="#cite_note-folkerts-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-burgimactutor_22-0" class="reference"><a href="#cite_note-burgimactutor-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p><a href="/w/index.php?title=Op%C4%87i_logaritam&action=edit&redlink=1" class="new" title="Opći logaritam (stranica ne postoji)">Opći logaritam</a> broja je indeks onog stepena od deset koji je jednak tom broju.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> Govoreći o broju koji zahtijeva mnogo cifara jeste grubi nagovještaj općeg logaritma, kojeg je spominjao <a href="/wiki/Arhimed" title="Arhimed">Arhimed</a> kao "red broja".<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Prvi realni logaritmi bile su heurističke metode koje su pretvarale množenje u sabiranje, čime se olakšava brzo računanje. Neke od tih metoda koristile su tablice izvedene iz trigonometrijskih identiteta.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Takve metoda se naziva <a href="/w/index.php?title=Prostafereza&action=edit&redlink=1" class="new" title="Prostafereza (stranica ne postoji)">prostafereza</a>. </p><p>Izum <a href="/wiki/Funkcija_(matematika)" title="Funkcija (matematika)">funkcije</a> sada poznate kao <a href="/wiki/Prirodni_logaritam" title="Prirodni logaritam">prirodni logaritam</a> počeo je kao pokušaj da se obavi <a href="/w/index.php?title=Kvadratura_(matematika)&action=edit&redlink=1" class="new" title="Kvadratura (matematika) (stranica ne postoji)">kvadratura</a> pravougaone <a href="/wiki/Hiperbola" title="Hiperbola">hiperbole</a> od <a href="/w/index.php?title=Gregoire_de_Saint_Vincent&action=edit&redlink=1" class="new" title="Gregoire de Saint Vincent (stranica ne postoji)">Gregoire de Saint Vincenta</a>, belgijskog Jezuita koji je boravio u Pragu. Arhimed je napisao <i>Kvadraturu hiperbole</i> u 3. vijeku p.n.e, ali kvadratura za hiperbolu izmicala je svima naporima dok Saint-Vincent nije objavio svoje rezultate 1647. godine. Veza koju pruža logaritam između <a href="/wiki/Geometrijska_progresija" title="Geometrijska progresija">geometrijske progresije</a> u svom <a href="/w/index.php?title=Argument_funkcije&action=edit&redlink=1" class="new" title="Argument funkcije (stranica ne postoji)">argumentu</a> i <a href="/w/index.php?title=Arimeti%C4%8Dka_progresija&action=edit&redlink=1" class="new" title="Arimetička progresija (stranica ne postoji)">aritmetičke progresije</a> vrijednosti, podstakla je <a href="/w/index.php?title=A._A._de_Sarasa&action=edit&redlink=1" class="new" title="A. A. de Sarasa (stranica ne postoji)">A. A. de Sarasa</a> da napravi vezu između Saint-Vincentove kvadrature i tradicije logaritama u prostaferezi, što je dovelo do pojma "hiperbolni logaritam", sinomnim za prirodni logaritam. Uskoro je nova funkcija cijenjena od strane naučnik: <a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygensa</a>, Pataviija, i <a href="/w/index.php?title=James_Gregory_(matemati%C4%8Dar)&action=edit&redlink=1" class="new" title="James Gregory (matematičar) (stranica ne postoji)">Jamesa Gregoryja</a>. Notaciju Log y je uveo <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> 1675. godine,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> a sljedeće godine on ju je povezao sa <a href="/wiki/Integral" title="Integral">integralom</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 \int {\frac {dy}{y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mi>y</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {dy}{y}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2507ed996fea98453b8d7bccdcd25cfc0295076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.435ex; height:5.843ex;" alt="{\displaystyle \int {\frac {dy}{y}}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Logaritamke_tablice,_logaritamska_skala_i_historijske_primjene"><span id="Logaritamke_tablice.2C_logaritamska_skala_i_historijske_primjene"></span>Logaritamke tablice, logaritamska skala i historijske primjene</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=10" title="Uredi odlomak "Logaritamke tablice, logaritamska skala i historijske primjene"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=10" title="Edit section's source code: Logaritamke tablice, logaritamska skala i historijske primjene"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Logarithms_Britannica_1797.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Logarithms_Britannica_1797.png/360px-Logarithms_Britannica_1797.png" decoding="async" width="360" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Logarithms_Britannica_1797.png/540px-Logarithms_Britannica_1797.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Logarithms_Britannica_1797.png/720px-Logarithms_Britannica_1797.png 2x" data-file-width="997" data-file-height="354" /></a><figcaption>Objašnjenje logaritma iz 1797. godine, <i><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a></i></figcaption></figure> <p>Pojednostavljenjem teških proračuna, logaritmi su doprinijeli razvoju nauke, naročito <a href="/wiki/Astronomija" title="Astronomija">astronomije</a>. Bili su značajni za napredak u <a href="/w/index.php?title=Anketa&action=edit&redlink=1" class="new" title="Anketa (stranica ne postoji)">anketiranje</a>, <a href="/w/index.php?title=Nebeska_navigacija&action=edit&redlink=1" class="new" title="Nebeska navigacija (stranica ne postoji)">nebeskoj navigaciji</a> i drugim domenama. <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> nazivao je logaritme: </p> <dl><dd><dl><dd>"...divljenja vrijedno lukavstvo koje, reduciranjem na nekoliko dana rad od nekoliko mjeseci, uduplava život astronoma, te ga pošteđuje grešaka i gađenja koje uzrokuje dugi proračun."<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></dd></dl></dd></dl> <p>Ključni alat koji je dopustio praktičnu upotrebu logaritama prije digitrona i računara bile su <i><a href="/w/index.php?title=Logaritamska_tablica&action=edit&redlink=1" class="new" title="Logaritamska tablica (stranica ne postoji)">logaritamske tablice</a></i>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Prvu takvu tablicu kompajlirao je <a href="/w/index.php?title=Henry_Briggs_(matemati%C4%8Dar)&action=edit&redlink=1" class="new" title="Henry Briggs (matematičar) (stranica ne postoji)">Henry Briggs</a> 1617. godine, odmah nakon Napierovog izuma. Naknadno, napravljene su tablice sa povećanim opsegom. Ove tablice su listale vrijednosti od log<sub><i>b</i></sub>(<i>x</i>) i <i>b</i><sup><i>x</i></sup> za svaki broj <i>x</i> u određenom opsegu, sa određenom preciznošću, za određenu bazu <i>b</i> (često <span style="white-space:nowrap"><i>b</i> = 10</span>). Naprimjer, Briggsova prva tabela sadržavala je opće logaritme svih cijelih brojeva u nizu 1–1000, sa preciznošću od 14 cifara. Kako je funkcija <span class="nowrap"><i>f</i>(<i>x</i>) = <i>b</i><sup><i>x</i></sup></span> inverzna funkcija od log<sub><i>b</i></sub>(<i>x</i>), bila je nazvana <b>antilogaritam</b>.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Proizvod i koeficijent od dva pozitivna broja <i>c</i> i <i>d</i> bili su rutinski računari kao suma i razlika njihovih logaritama. Proizvod <i>cd</i> ili koeficijent <i>c</i>/<i>d</i> dolazio je od uzimanja antilogaritma zbira ili razlike, također preko iste tabele: </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 cd=b^{\log _{b}(c)}\,b^{\log _{b}(d)}=b^{\log _{b}(c)+\log _{b}(d)}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>d</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cd=b^{\log _{b}(c)}\,b^{\log _{b}(d)}=b^{\log _{b}(c)+\log _{b}(d)}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579a1d590c4c91b1578fc2647feea937ddf71202" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:33.776ex; height:2.843ex;" alt="{\displaystyle cd=b^{\log _{b}(c)}\,b^{\log _{b}(d)}=b^{\log _{b}(c)+\log _{b}(d)}\,}"></span></dd></dl> <p>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 {\frac {c}{d}}=cd^{-1}=b^{\log _{b}(c)-\log _{b}(d)}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>d</mi> </mfrac> </mrow> <mo>=</mo> <mi>c</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {c}{d}}=cd^{-1}=b^{\log _{b}(c)-\log _{b}(d)}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6b1fd6ff8aa7afecfa4c74006f53623aee9f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.156ex; height:4.843ex;" alt="{\displaystyle {\frac {c}{d}}=cd^{-1}=b^{\log _{b}(c)-\log _{b}(d)}.\,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Analitička_svojstva"><span id="Analiti.C4.8Dka_svojstva"></span>Analitička svojstva</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=11" title="Uredi odlomak "Analitička svojstva"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=11" title="Edit section's source code: Analitička svojstva"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dublji studiji logaritama zahtijevaju koncept <i><a href="/wiki/Funkcija_(matematika)" title="Funkcija (matematika)">funkcije</a></i>. Funkcija je pravilo koje, kada mu se da broj, proizvodi neki drugi broj.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> Primjer je funkcija koja proizvodi <span class="nowrap"><i>x</i>-ti</span> stepen od <i>b</i> za bilo koji realan broj <i>x</i>, gdje je baza <i>b</i> fiksni broj. Ova se funkcija piše kao </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 f(x)=b^{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> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=b^{x}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9592c4d49d061f3e4e899ba7473b40d1863be38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.72ex; height:2.843ex;" alt="{\displaystyle f(x)=b^{x}.\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Logaritamska_funkcija">Logaritamska funkcija</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=12" title="Uredi odlomak "Logaritamska funkcija"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=12" title="Edit section's source code: Logaritamska funkcija"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Da bi se opravdala definicija logaritama, potrebno je pokazati da jednačina </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{x}=y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{x}=y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c648d30dc71f430a3d3983633e248e4b5afe7af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.811ex; height:2.676ex;" alt="{\displaystyle b^{x}=y\,}"></span></dd></dl> <p>ima rješenje <i>x</i> i da je rješenje jedinstveno, pod uvjetom da je <i>y</i> pozitivan i da je <i>b</i> pozitivan i različit od 1. Dokaz ovog slučaja zahtijeva teoremu o srednjoj vrijednosti iz elementarnog <a href="/wiki/Kalkulus" class="mw-redirect" title="Kalkulus">kalkulusa</a>.<sup id="cite_ref-LangIII.3_31-0" class="reference"><a href="#cite_note-LangIII.3-31"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Ova teorema drži da <a href="/wiki/Neprekidna_funkcija" title="Neprekidna funkcija">neprekidna funkcija</a> koja proizvodi dvije vrijednosti <i>m</i> i <i>n</i> također proizvodi bilo koju vrijednost koja leži između <i>m</i> i <i>n</i>. Funkcija je <i>neprekidna</i> ako ne "skače", tj. ako se njezin grafik može nacrtati bez podizanja olovke. </p><p>Ovo svojestvo može biti prikazano da važi za funkciju <span style="white-space:nowrap"><i>f</i>(<i>x</i>) = <i>b</i><sup><i>x</i></sup></span>. Pošto <i>f</i> uzima proizvoljno velike i proizvoljno male pozitivne vrijednosti, bilo koji broj <span class="nowrap"><i>y</i> > 0</span> leži između <i>f</i>(<i>x</i><sub>0</sub>) i <i>f</i>(<i>x</i><sub>1</sub>) za odgovarajući <i>x</i><sub>0</sub> and <i>x</i><sub>1</sub>. Stoga, teorema o srednjoj vrijednosti osigurava da jednačina <i>f</i>(<i>x</i>) = <i>y</i> ima rješenje. Štaviše, postoji samo jedno rješene za ovu jednačinu, jer je funkcija <i>f</i> <a href="/w/index.php?title=Monotona_funkcija&action=edit&redlink=1" class="new" title="Monotona funkcija (stranica ne postoji)">strogo rastuća</a> (za <span class="nowrap"><i>b</i> > 1</span>), ili strogo opadajuća (za <span class="nowrap">0 < <i>b</i> < 1</span>).<sup id="cite_ref-LangIV.2_32-0" class="reference"><a href="#cite_note-LangIV.2-32"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>Jedinstveno rješenje <i>x</i> je logaritam od <i>y</i> za bazu <i>b</i>, log<sub><i>b</i></sub>(<i>y</i>). Funkcija koja dodjeljuje <i>y</i> svoj logaritam zove se <i>logaritamska funkcija</i> ili <i>logaritmična funkcija</i> (ili samo <i>logaritam</i>). </p><p>Funkcija log<sub><i>b</i></sub>(<i>x</i>) je u suštini okarakerisana formulom proizvoda iznad </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8808e1aeb29ec9cdd4f91db22622bbc345b1564f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.712ex; height:2.843ex;" alt="{\displaystyle \log _{b}(xy)=\log _{b}(x)+\log _{b}(y).}"></span></dd></dl> <p>Preciznije, logaritam za svaku bazu <span class="nowrap"><i>b</i> > 1</span> je samo <a href="/w/index.php?title=Rastu%C4%87a_funkcija&action=edit&redlink=1" class="new" title="Rastuća funkcija (stranica ne postoji)">rastuća funkcija</a> <i>f</i> od pozitivnih realnih brojeva do realnih brojeva koji zadovoljavaju <span style="white-space:nowrap"><i>f</i>(<i>b</i>) = 1</span> i <sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </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 f(xy)=f(x)+f(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(xy)=f(x)+f(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee6f0eb6e355d16f673a0d4a21705e24a227008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.82ex; height:2.843ex;" alt="{\displaystyle f(xy)=f(x)+f(y).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Također_pogledajte"><span id="Tako.C4.91er_pogledajte"></span>Također pogledajte</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=13" title="Uredi odlomak "Također pogledajte"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=13" title="Edit section's source code: Također pogledajte"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Prirodni_logaritam" title="Prirodni logaritam">Prirodni logaritam</a></li> <li><a href="/w/index.php?title=Beskona%C4%8Dni_logaritam&action=edit&redlink=1" class="new" title="Beskonačni logaritam (stranica ne postoji)">Beskonačni logaritam</a></li> <li><a href="/w/index.php?title=Iterativni_logaritam&action=edit&redlink=1" class="new" title="Iterativni logaritam (stranica ne postoji)">Iterativni logaritam</a></li> <li><a href="/w/index.php?title=Diskretni_logaritam&action=edit&redlink=1" class="new" title="Diskretni logaritam (stranica ne postoji)">Diskretni logaritam</a></li> <li><a href="/w/index.php?title=Zechovi_logaritmi&action=edit&redlink=1" class="new" title="Zechovi logaritmi (stranica ne postoji)">Zechovi logaritmi</a></li> <li><a href="/w/index.php?title=Logaritam_matrice&action=edit&redlink=1" class="new" title="Logaritam matrice (stranica ne postoji)">Logaritam matrice</a></li> <li><a href="/w/index.php?title=Decibel&action=edit&redlink=1" class="new" title="Decibel (stranica ne postoji)">Decibel</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Bilješke"><span id="Bilje.C5.A1ke"></span>Bilješke</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=14" title="Uredi odlomak "Bilješke"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=14" title="Edit section's source code: Bilješke"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3312388">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Restrikcije na <i>x</i> i <i>b</i> su opisane u sekciji <a href="#Analitička_svojstva">"Analitička svojstva"</a>.</span> </li> <li id="cite_note-adaa-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-adaa_16-0">^</a></b></span> <span class="reference-text">Neki matematičari ne podržavaju ovu notaciju. U njegovoj autobiografiji iz 1985., <a href="/w/index.php?title=Paul_Halmos&action=edit&redlink=1" class="new" title="Paul Halmos (stranica ne postoji)">Paul Halmos</a> je kritizirao ono što je smatrao "dječija ln notacija", za koju je rekao da je nijedan matematičar nikad nije koristio.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Notaciju je uveo <a href="/w/index.php?title=Irving_Stringham&action=edit&redlink=1" class="new" title="Irving Stringham (stranica ne postoji)">Irving Stringham</a>, matematičar.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Naprimjer <a href="/wiki/C_(programski_jezik)" title="C (programski jezik)">C</a>, <a href="/wiki/Java_(programski_jezik)" title="Java (programski jezik)">Java</a>, <a href="/w/index.php?title=Haskell_(programski_jezik)&action=edit&redlink=1" class="new" title="Haskell (programski jezik) (stranica ne postoji)">Haskell</a> i <a href="/wiki/BASIC" title="BASIC">BASIC</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Reference">Reference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=15" title="Uredi odlomak "Reference"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=15" title="Edit section's source code: Reference"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3312388"><div class="reflist reflist-columns references-column-width reflist-columns-2" style=""> <ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r3303350">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFKateBhapkar2009" class="citation cs2">Kate, S.K.; Bhapkar, H.R. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=v4R0GSJtEQ4C&pg=PR1#v=onepage&q&f=false"><i>Basics Of Mathematics</i></a>, Pune: Technical Publications, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-81-8431-755-8" title="Posebno:Knjižni izvori/978-81-8431-755-8"><bdi>978-81-8431-755-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basics+Of+Mathematics&rft.place=Pune&rft.pub=Technical+Publications&rft.date=2009&rft.isbn=978-81-8431-755-8&rft.aulast=Kate&rft.aufirst=S.K.&rft.au=Bhapkar%2C+H.R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dv4R0GSJtEQ4C%26pg%3DPR1%23v%3Donepage%26q%26f%3Dfalse&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span>, poglavlje 1</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Svi iskazi u ovoj sekciji mogu biti pronađeni u Shailesh Shirali <a href="#CITEREFShirali2002">2002</a>, section 4, (Douglas Downing <a href="#CITEREFDowning2003">2003</a>, p. 275), ili Kate & Bhapkar <a href="#CITEREFKateBhapkar2009">2009</a>, p. 1-1, naprimjer.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFBernsteinBernstein1999" class="citation cs2">Bernstein, Stephen; Bernstein, Ruth (1999), <span class="cs1-lock-registration" title="Potrebna besplatna registracija"><a rel="nofollow" class="external text" href="https://archive.org/details/schaumsoutlineof00bern"><i>Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probability</i></a></span>, Schaum's outline series, New York: <a href="/w/index.php?title=McGraw-Hill&action=edit&redlink=1" class="new" title="McGraw-Hill (stranica ne postoji)">McGraw-Hill</a>, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-07-005023-5" title="Posebno:Knjižni izvori/978-0-07-005023-5"><bdi>978-0-07-005023-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+outline+of+theory+and+problems+of+elements+of+statistics.+I%2C+Descriptive+statistics+and+probability&rft.place=New+York&rft.series=Schaum%27s+outline+series&rft.pub=McGraw-Hill&rft.date=1999&rft.isbn=978-0-07-005023-5&rft.aulast=Bernstein&rft.aufirst=Stephen&rft.au=Bernstein%2C+Ruth&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fschaumsoutlineof00bern&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span>, str. 21</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFDowning2003" class="citation cs2">Downing, Douglas (2003), <span class="cs1-lock-registration" title="Potrebna besplatna registracija"><a rel="nofollow" class="external text" href="https://archive.org/details/algebraeasyway00down_0"><i>Algebra the Easy Way</i></a></span>, Barron's Educational Series, Hauppauge, N.Y.: Barron's, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-7641-1972-9" title="Posebno:Knjižni izvori/978-0-7641-1972-9"><bdi>978-0-7641-1972-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+the+Easy+Way&rft.place=Hauppauge%2C+N.Y.&rft.series=Barron%27s+Educational+Series&rft.pub=Barron%27s&rft.date=2003&rft.isbn=978-0-7641-1972-9&rft.aulast=Downing&rft.aufirst=Douglas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebraeasyway00down_0&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span>, chapter 17, str. 275</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFWegener2005" class="citation cs2">Wegener, Ingo (2005), <i>Complexity theory: exploring the limits of efficient algorithms</i>, Berlin, New York: <a href="/w/index.php?title=Springer-Verlag&action=edit&redlink=1" class="new" title="Springer-Verlag (stranica ne postoji)">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-3-540-21045-0" title="Posebno:Knjižni izvori/978-3-540-21045-0"><bdi>978-3-540-21045-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Complexity+theory%3A+exploring+the+limits+of+efficient+algorithms&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=2005&rft.isbn=978-3-540-21045-0&rft.aulast=Wegener&rft.aufirst=Ingo&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span>, str. 20</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFVan_der_Lubbe1997" class="citation cs2">Van der Lubbe, Jan C. A. (1997), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tBuI_6MQTcwC&pg=PA3"><i>Information Theory</i></a>, Cambridge University Press, str. 3, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/9780521467605" title="Posebno:Knjižni izvori/9780521467605"><bdi>9780521467605</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Information+Theory&rft.pages=3&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=9780521467605&rft.aulast=Van+der+Lubbe&rft.aufirst=Jan+C.+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtBuI_6MQTcwC%26pg%3DPA3&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFAllenTriantaphillidou2011" class="citation cs2">Allen, Elizabeth; Triantaphillidou, Sophie (2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IfWivY3mIgAC&pg=PA228"><i>The Manual of Photography</i></a>, Taylor & Francis, str. 228, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/9780240520377" title="Posebno:Knjižni izvori/9780240520377"><bdi>9780240520377</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Manual+of+Photography&rft.pages=228&rft.pub=Taylor+%26+Francis&rft.date=2011&rft.isbn=9780240520377&rft.aulast=Allen&rft.aufirst=Elizabeth&rft.au=Triantaphillidou%2C+Sophie&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIfWivY3mIgAC%26pg%3DPA228&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFFranz_EmbacherPetra_Oberhuemer" class="citation cs2">Franz Embacher; Petra Oberhuemer, <a rel="nofollow" class="external text" href="http://www.mathe-online.at/mathint/lexikon/l.html"><i>Mathematisches Lexikon</i></a> (jezik: njemački), mathe online: für Schule, Fachhochschule, Universität unde Selbststudium<span class="reference-accessdate">, pristupljeno 22. 3. 2011</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematisches+Lexikon&rft.pub=mathe+online%3A+f%C3%BCr+Schule%2C+Fachhochschule%2C+Universit%C3%A4t+unde+Selbststudium&rft.au=Franz+Embacher&rft.au=Petra+Oberhuemer&rft_id=http%3A%2F%2Fwww.mathe-online.at%2Fmathint%2Flexikon%2Fl.html&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFTaylor1995" class="citation cs2">Taylor, B. N. (1995), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070629210131/http://physics.nist.gov/Pubs/SP811/sec10.html#10.1.2"><i>Guide for the Use of the International System of Units (SI)</i></a>, US Department of Commerce, arhivirano s <a rel="nofollow" class="external text" href="http://physics.nist.gov/Pubs/SP811/sec10.html#10.1.2">originala</a>, 29. 6. 2007<span class="reference-accessdate">, pristupljeno 27. 7. 2016</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Guide+for+the+Use+of+the+International+System+of+Units+%28SI%29&rft.pub=US+Department+of+Commerce&rft.date=1995&rft.aulast=Taylor&rft.aufirst=B.+N.&rft_id=http%3A%2F%2Fphysics.nist.gov%2FPubs%2FSP811%2Fsec10.html%2310.1.2&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-gullberg-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-gullberg_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFGullberg,_Jan1997" class="citation cs2">Gullberg, Jan (1997), <span class="cs1-lock-registration" title="Potrebna besplatna registracija"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsfromb1997gull"><i>Mathematics: from the birth of numbers.</i></a></span>, New York: W. W. Norton & Co, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-393-04002-9" title="Posebno:Knjižni izvori/978-0-393-04002-9"><bdi>978-0-393-04002-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%3A+from+the+birth+of+numbers.&rft.place=New+York&rft.pub=W.+W.+Norton+%26+Co&rft.date=1997&rft.isbn=978-0-393-04002-9&rft.au=Gullberg%2C+Jan&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsfromb1997gull&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Pogledati fusnotu 1 u <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFPerlReingold1977" class="citation journal cs1">Perl, Yehoshua; Reingold, Edward M. (decembar 1977). "Understanding the complexity of interpolation search". <i>Information Processing Letters</i>. <b>6</b> (6): 219–222. <a href="/wiki/Doi_(identifikator)" class="mw-redirect" title="Doi (identifikator)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0020-0190%2877%2990072-2">10.1016/0020-0190(77)90072-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+Processing+Letters&rft.atitle=Understanding+the+complexity+of+interpolation+search&rft.volume=6&rft.issue=6&rft.pages=219-222&rft.date=1977-12&rft_id=info%3Adoi%2F10.1016%2F0020-0190%2877%2990072-2&rft.aulast=Perl&rft.aufirst=Yehoshua&rft.au=Reingold%2C+Edward+M.&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFPaul_Halmos1985" class="citation cs2">Paul Halmos (1985), <i>I Want to Be a Mathematician: An Automathography</i>, Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-387-96078-4" title="Posebno:Knjižni izvori/978-0-387-96078-4"><bdi>978-0-387-96078-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=I+Want+to+Be+a+Mathematician%3A+An+Automathography&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1985&rft.isbn=978-0-387-96078-4&rft.au=Paul+Halmos&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFIrving_Stringham1893" class="citation cs2">Irving Stringham (1893), <a rel="nofollow" class="external text" href="https://books.google.com/?id=hPEKAQAAIAAJ&pg=PR13&dq=%22Irving+Stringham%22+In-natural-logarithm&q="><i>Uniplanar algebra: being part I of a propædeutic to the higher mathematical analysis</i></a>, The Berkeley Press, str. xiii</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Uniplanar+algebra%3A+being+part+I+of+a+prop%C3%A6deutic+to+the+higher+mathematical+analysis&rft.pages=xiii&rft.pub=The+Berkeley+Press&rft.date=1893&rft.au=Irving+Stringham&rft_id=https%3A%2F%2Fbooks.google.com%2F%3Fid%3DhPEKAQAAIAAJ%26pg%3DPR13%26dq%3D%2522Irving%2BStringham%2522%2BIn-natural-logarithm%26q%3D&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFRoy_S._Freedman2006" class="citation cs2">Roy S. Freedman (2006), <a rel="nofollow" class="external text" href="https://books.google.com/?id=APJ7QeR_XPkC&pg=PA59"><i>Introduction to Financial Technology</i></a>, Amsterdam: Academic Press, str. 59, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-12-370478-8" title="Posebno:Knjižni izvori/978-0-12-370478-8"><bdi>978-0-12-370478-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Financial+Technology&rft.place=Amsterdam&rft.pages=59&rft.pub=Academic+Press&rft.date=2006&rft.isbn=978-0-12-370478-8&rft.au=Roy+S.+Freedman&rft_id=https%3A%2F%2Fbooks.google.com%2F%3Fid%3DAPJ7QeR_XPkC%26pg%3DPA59&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">vidjeti teoremu 3.29 u <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFRudin1984" class="citation book cs1">Rudin, Walter (1984). <span class="cs1-lock-registration" title="Potrebna besplatna registracija"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi"><i>Principles of mathematical analysis</i></a></span> (3rd ed., International student izd.). Auckland: McGraw-Hill International. <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0070856134" title="Posebno:Knjižni izvori/978-0070856134"><bdi>978-0070856134</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+mathematical+analysis&rft.place=Auckland&rft.edition=3rd+ed.%2C+International+student&rft.pub=McGraw-Hill+International&rft.date=1984&rft.isbn=978-0070856134&rft.aulast=Rudin&rft.aufirst=Walter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesofmath00rudi&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFNapier1614" class="citation cs2"><a href="/wiki/John_Napier" title="John Napier">Napier, John</a> (1614), <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN527914568&DMDID=DMDLOG_0001&LOGID=LOG_0001&PHYSID=PHYS_0001"><i>Mirifici Logarithmorum Canonis Descriptio</i></a> [<i>The Description of the Wonderful Rule of Logarithms</i>] (jezik: Latin), Edinburgh, Scotland: Andrew Hart</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mirifici+Logarithmorum+Canonis+Descriptio&rft.place=Edinburgh%2C+Scotland&rft.pub=Andrew+Hart&rft.date=1614&rft.aulast=Napier&rft.aufirst=John&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdms%2Fload%2Fimg%2F%3FPPN%3DPPN527914568%26DMDID%3DDMDLOG_0001%26LOGID%3DLOG_0001%26PHYSID%3DPHYS_0001&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span><span class="cs1-maint citation-comment">CS1 održavanje: nepoznati jezik (<a href="/wiki/Kategorija:CS1_odr%C5%BEavanje:_nepoznati_jezik" title="Kategorija:CS1 održavanje: nepoznati jezik">link</a>)</span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFHobson1914" class="citation cs2">Hobson, Ernest William (1914), <a rel="nofollow" class="external text" href="https://archive.org/details/johnnapierinvent00hobsiala"><i>John Napier and the invention of logarithms, 1614</i></a>, Cambridge: The University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=John+Napier+and+the+invention+of+logarithms%2C+1614&rft.place=Cambridge&rft.pub=The+University+Press&rft.date=1914&rft.aulast=Hobson&rft.aufirst=Ernest+William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fjohnnapierinvent00hobsiala&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-folkerts-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-folkerts_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFFolkertsLaunertThom" class="citation cs2">Folkerts, Menso; Launert, Dieter; Thom, Andreas (otobar 2015), <i>Jost Bürgi's Method for Calculating Sines</i>, <a href="/w/index.php?title=ArXiv_(identifikator)&action=edit&redlink=1" class="new" title="ArXiv (identifikator) (stranica ne postoji)">arXiv</a>:<span class="cs1-lock-free" title="Slobodno dostupno"><a rel="nofollow" class="external text" href="//arxiv.org/abs/1510.03180">1510.03180</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Jost+B%C3%BCrgi%27s+Method+for+Calculating+Sines&rft_id=info%3Aarxiv%2F1510.03180&rft.aulast=Folkerts&rft.aufirst=Menso&rft.au=Launert%2C+Dieter&rft.au=Thom%2C+Andreas&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span> <span class="cs1-visible-error error citation-comment">Provjerite vrijednost datuma u parametru: <code class="cs1-code">|date=</code> (<a href="/w/index.php?title=Pomo%C4%87:CS1_gre%C5%A1ke&action=edit&redlink=1" class="new" title="Pomoć:CS1 greške (stranica ne postoji)">pomoć</a>)</span></span> </li> <li id="cite_note-burgimactutor-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-burgimactutor_22-0">^</a></b></span> <span class="reference-text">MacTutor članak @ Jost Bürgi: <a rel="nofollow" class="external free" href="http://www-history.mcs.st-and.ac.uk/Biographies/Burgi.html">http://www-history.mcs.st-and.ac.uk/Biographies/Burgi.html</a></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">William Gardner (1742) <i>Tables of Logarithms</i></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">R.C. Pierce (1977) "A brief history of logarithm", <a href="/w/index.php?title=Two-Year_College_Mathematics_Journal&action=edit&redlink=1" class="new" title="Two-Year College Mathematics Journal (stranica ne postoji)">Two-Year College Mathematics Journal</a> 8(1):22–6.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Enrique Gonzales-Velasco (2011) <i>Journey through Mathematics – Creative Episodes in its History</i>, §2.4 Hyperbolic logarithms, str. 117, Springer <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-387-92153-2" title="Posebno:Knjižni izvori/978-0-387-92153-2">978-0-387-92153-2</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><a href="/w/index.php?title=Florian_Cajori&action=edit&redlink=1" class="new" title="Florian Cajori (stranica ne postoji)">Florian Cajori</a> (1913) "History of the exponential and logarithm concepts", <a href="/w/index.php?title=American_Mathematical_Monthly&action=edit&redlink=1" class="new" title="American Mathematical Monthly (stranica ne postoji)">American Mathematical Monthly</a> 20: 5, 35, 75, 107, 148, 173, 205.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFBryant" class="citation cs2">Bryant, Walter W., <a rel="nofollow" class="external text" href="https://archive.org/stream/ahistoryastrono01bryagoog#page/n72/mode/2up"><i>A History of Astronomy</i></a>, London: Methuen & Co</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Astronomy&rft.place=London&rft.pub=Methuen+%26+Co&rft.aulast=Bryant&rft.aufirst=Walter+W.&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fahistoryastrono01bryagoog%23page%2Fn72%2Fmode%2F2up&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span>, str. 44</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFCampbell-Kelly2003" class="citation cs2">Campbell-Kelly, Martin (2003), <i>The history of mathematical tables: from Sumer to spreadsheets</i>, Oxford scholarship online, <a href="/w/index.php?title=Oxford_University_Press&action=edit&redlink=1" class="new" title="Oxford University Press (stranica ne postoji)">Oxford University Press</a>, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-19-850841-0" title="Posebno:Knjižni izvori/978-0-19-850841-0"><bdi>978-0-19-850841-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+history+of+mathematical+tables%3A+from+Sumer+to+spreadsheets&rft.series=Oxford+scholarship+online&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=978-0-19-850841-0&rft.aulast=Campbell-Kelly&rft.aufirst=Martin&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span>, sekcija 2</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFAbramowitzStegun1972" class="citation cs2"><a href="/w/index.php?title=Milton_Abramowitz&action=edit&redlink=1" class="new" title="Milton Abramowitz (stranica ne postoji)">Abramowitz, Milton</a>; <a href="/w/index.php?title=Irene_Stegun&action=edit&redlink=1" class="new" title="Irene Stegun (stranica ne postoji)">Stegun, Irene A.</a>, ured. (1972), <i><a href="/w/index.php?title=Handbook_of_Mathematical_Functions_with_Formulas,_Graphs,_and_Mathematical_Tables&action=edit&redlink=1" class="new" title="Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (stranica ne postoji)">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a></i> (10th izd.), New York: <a href="/w/index.php?title=Dover_Publications&action=edit&redlink=1" class="new" title="Dover Publications (stranica ne postoji)">Dover Publications</a>, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-486-61272-0" title="Posebno:Knjižni izvori/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&rft.place=New+York&rft.edition=10th&rft.pub=Dover+Publications&rft.date=1972&rft.isbn=978-0-486-61272-0&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span>, sekcija 4.7., str. 89</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFDevlin2004" class="citation book cs1"><a href="/w/index.php?title=Keith_Devlin&action=edit&redlink=1" class="new" title="Keith Devlin (stranica ne postoji)">Devlin, Keith</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uQHF7bcm4k4C&printsec=frontcover#v=onepage&q&f=false"><i>Sets, functions, and logic: an introduction to abstract mathematics</i></a>. Chapman & Hall/CRC mathematics (3rd izd.). Boca Raton, Fla: Chapman & Hall/CRC. <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/1-58488-449-5" title="Posebno:Knjižni izvori/1-58488-449-5"><bdi>1-58488-449-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets%2C+functions%2C+and+logic%3A+an+introduction+to+abstract+mathematics&rft.place=Boca+Raton%2C+Fla&rft.series=Chapman+%26+Hall%2FCRC+mathematics&rft.edition=3rd&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2004&rft.isbn=1-58488-449-5&rft.aulast=Devlin&rft.aufirst=Keith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuQHF7bcm4k4C%26printsec%3Dfrontcover%23v%3Donepage%26q%26f%3Dfalse&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></span> </li> <li id="cite_note-LangIII.3-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-LangIII.3_31-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFLang1997" class="citation cs2"><a href="/w/index.php?title=Serge_Lang&action=edit&redlink=1" class="new" title="Serge Lang (stranica ne postoji)">Lang, Serge</a> (1997), <i>Undergraduate analysis</i>, <a href="/w/index.php?title=Undergraduate_Texts_in_Mathematics&action=edit&redlink=1" class="new" title="Undergraduate Texts in Mathematics (stranica ne postoji)">Undergraduate Texts in Mathematics</a> (2nd izd.), Berlin, New York: <a href="/w/index.php?title=Springer-Verlag&action=edit&redlink=1" class="new" title="Springer-Verlag (stranica ne postoji)">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-0-387-94841-6" title="Posebno:Knjižni izvori/978-0-387-94841-6"><bdi>978-0-387-94841-6</bdi></a>, <a href="/w/index.php?title=MR_(identifikator)&action=edit&redlink=1" class="new" title="MR (identifikator) (stranica ne postoji)">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1476913">1476913</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Undergraduate+analysis&rft.place=Berlin%2C+New+York&rft.series=Undergraduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=978-0-387-94841-6&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1476913%23id-name%3DMR&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span>, sekcija III.3</span> </li> <li id="cite_note-LangIV.2-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-LangIV.2_32-0">^</a></b></span> <span class="reference-text">Lang <a href="#CITEREFLang1997">1997</a>, section IV.2</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFDieudonné1969" class="citation book cs1">Dieudonné, Jean (1969). <i>Foundations of Modern Analysis</i>. <b>1</b>. Academic Press. str. 84.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Modern+Analysis&rft.pages=84&rft.pub=Academic+Press&rft.date=1969&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span> item (4.3.1)</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Vanjski_linkovi">Vanjski linkovi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logaritam&veaction=edit&section=16" title="Uredi odlomak "Vanjski linkovi"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logaritam&action=edit&section=16" title="Edit section's source code: Vanjski linkovi"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li class="mw-empty-elt"></li></ul> <style data-mw-deduplicate="TemplateStyles:r3627399">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box metadata side-box-right"><style data-mw-deduplicate="TemplateStyles:r3486241">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Datoteka:Commons-logo.svg" class="mw-file-description" title="Commons logo"><img alt="Commons logo" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/60px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a><figcaption>Commons logo</figcaption></figure></div> <div class="side-box-text plainlist"><a href="https://commons.wikimedia.org/wiki/Po%C4%8Detna_strana" class="extiw" title="commons:Početna strana">Commons</a> ima datoteke na temu: <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Logarithm?uselang=bs">Logaritam </a></span></b></div></div> </div> <ul><li class="mw-empty-elt"></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3627399"><div class="side-box metadata side-box-right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3486241"> <div class="side-box-flex"> <div class="side-box-image"><figure class="mw-halign-none" typeof="mw:File"><a href="/wiki/Datoteka:Wiktionary-logo.svg" class="mw-file-description" title="Logo Wikirječnika"><img alt="Logo Wikirječnika" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Wiktionary-logo.svg/30px-Wiktionary-logo.svg.png" decoding="async" width="30" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Wiktionary-logo.svg/45px-Wiktionary-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Wiktionary-logo.svg/60px-Wiktionary-logo.svg.png 2x" data-file-width="370" data-file-height="350" /></a><figcaption>Logo Wikirječnika</figcaption></figure></div> <div class="side-box-text plainlist">Potražite <i><b><a href="https://bs.wiktionary.org/wiki/Logarithm" class="extiw" title="wikt:Logarithm">Logarithm</a></b></i> na <a href="https://bs.wiktionary.org/wiki/Po%C4%8Detna_strana" class="extiw" title="wikt:Početna strana">Wikirječniku</a>, slobodnom rječniku.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20121218200616/http://www.khanacademy.org/math/algebra/logarithms-tutorial">Khan Academy: Logarithms, free online micro lectures</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFHazewinkel2001" class="citation cs2">Hazewinkel, Michiel, ured. (2001), <a rel="nofollow" class="external text" href="http://eom.springer.de/p/l060600.htm">"Logarithmic function"</a>, <i><a href="/w/index.php?title=Matemati%C4%8Dka_enciklopedija&action=edit&redlink=1" class="new" title="Matematička enciklopedija (stranica ne postoji)">Matematička enciklopedija</a></i>, Kluwer Academic Publishers, <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:Knji%C5%BEni_izvori/978-1556080104" title="Posebno:Knjižni izvori/978-1556080104"><bdi>978-1556080104</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Logarithmic+function&rft.btitle=Matemati%C4%8Dka+enciklopedija&rft.pub=Kluwer+Academic+Publishers&rft.date=2001&rft.isbn=978-1556080104&rft_id=http%3A%2F%2Feom.springer.de%2Fp%2Fl060600.htm&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFColin_Byfleet" class="citation cs2">Colin Byfleet, <a rel="nofollow" class="external text" href="http://mediasite.oddl.fsu.edu/mediasite/Viewer/?peid=003298f9a02f468c8351c50488d6c479"><i>Educational video on logarithms</i></a><span class="reference-accessdate">, pristupljeno 12. 10. 2010</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Educational+video+on+logarithms&rft.au=Colin+Byfleet&rft_id=http%3A%2F%2Fmediasite.oddl.fsu.edu%2Fmediasite%2FViewer%2F%3Fpeid%3D003298f9a02f468c8351c50488d6c479&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r3303350"><cite id="CITEREFEdward_Wright" class="citation cs2">Edward Wright, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20021203005508/http://www.johnnapier.com/table_of_logarithms_001.htm"><i>Translation of Napier's work on logarithms</i></a>, arhivirano s <a rel="nofollow" class="external text" href="http://www.johnnapier.com/table_of_logarithms_001.htm">originala</a>, 3. 12. 2002<span class="reference-accessdate">, pristupljeno 12. 10. 2010</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Translation+of+Napier%27s+work+on+logarithms&rft.au=Edward+Wright&rft_id=http%3A%2F%2Fwww.johnnapier.com%2Ftable_of_logarithms_001.htm&rfr_id=info%3Asid%2Fbs.wikipedia.org%3ALogaritam" class="Z3988"></span></li></ul> <div role="navigation" class="navbox" aria-labelledby="Elementarna_aritmetika" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style 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