对数 - 维基百科，自由的百科全书

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href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E5%AF%B9%E6%95%B0" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E5%AF%B9%E6%95%B0" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-定义" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#定义"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>定义</span> </div> </a> <ul id="toc-定义-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-历史" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#历史"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>历史</span> </div> </a> <button aria-controls="toc-历史-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>开关历史子章节</span> </button> <ul id="toc-历史-sublist" class="vector-toc-list"> <li id="toc-对数" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#对数"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>对数</span> </div> </a> <ul id="toc-对数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-符号" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#符号"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>符号</span> </div> </a> <ul id="toc-符号-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-对数函数" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#对数函数"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>对数函数</span> </div> </a> <ul id="toc-对数函数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-运算公式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#运算公式"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>运算公式</span> </div> </a> <ul id="toc-运算公式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-有理和无理指数" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#有理和无理指数"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>有理和无理指数</span> </div> </a> <ul id="toc-有理和无理指数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-特殊底数" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#特殊底数"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>特殊底数</span> </div> </a> <ul id="toc-特殊底数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-底数变换" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#底数变换"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>底数变换</span> </div> </a> <ul id="toc-底数变换-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-对数的用途" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#对数的用途"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>对数的用途</span> </div> </a> <button aria-controls="toc-对数的用途-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>开关对数的用途子章节</span> </button> <ul id="toc-对数的用途-sublist" class="vector-toc-list"> <li id="toc-简便计算" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#简便计算"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>简便计算</span> </div> </a> <ul id="toc-简便计算-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-群论" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#群论"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>群论</span> </div> </a> <ul id="toc-群论-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-复对数" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#复对数"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>复对数</span> </div> </a> <ul id="toc-复对数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-微积分" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#微积分"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>微积分</span> </div> </a> <ul id="toc-微积分-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-计算自然对数的级数" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#计算自然对数的级数"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>计算自然对数的级数</span> </div> </a> <ul id="toc-计算自然对数的级数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-计算机" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#计算机"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>计算机</span> </div> </a> <ul id="toc-计算机-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-一般化" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#一般化"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>一般化</span> </div> </a> <ul id="toc-一般化-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-对数表" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#对数表"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>对数表</span> </div> </a> <ul id="toc-对数表-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-注释" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#注释"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>注释</span> </div> </a> <ul id="toc-注释-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">18</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> <span class="vector-toc-numb">19</span> <span>外部链接</span> </div> </a> <ul id="toc-外部链接-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="目录" 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="开关目录" > <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">开关目录</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">对数</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="前往另一种语言写成的文章。109种语言可用" > <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种语言</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 – 南非荷兰语" lang="af" hreflang="af" data-title="Logaritme" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" 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 – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Logarithmus" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" 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 – 阿姆哈拉语" lang="am" hreflang="am" data-title="ሎጋሪዝም" data-language-autonym="አማርኛ" data-language-local-name="阿姆哈拉语" 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 – 阿拉贡语" lang="an" hreflang="an" data-title="Logaritmo" data-language-autonym="Aragonés" data-language-local-name="阿拉贡语" 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 – 阿拉伯语" lang="ar" hreflang="ar" data-title="لوغاريتم" data-language-autonym="العربية" data-language-local-name="阿拉伯语" 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 – 摩洛哥阿拉伯文" lang="ary" hreflang="ary" data-title="لوڭاريتم" data-language-autonym="الدارجة" data-language-local-name="摩洛哥阿拉伯文" 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 – 阿萨姆语" lang="as" hreflang="as" data-title="ঘাতাংক" data-language-autonym="অসমীয়া" data-language-local-name="阿萨姆语" 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 – 阿斯图里亚斯语" lang="ast" hreflang="ast" data-title="Logaritmu" data-language-autonym="Asturianu" data-language-local-name="阿斯图里亚斯语" 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 – 阿塞拜疆语" lang="az" hreflang="az" data-title="Loqarifm" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" 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 – 巴什基尔语" lang="ba" hreflang="ba" data-title="Логарифм" data-language-autonym="Башҡортса" data-language-local-name="巴什基尔语" 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 – 薩莫吉希亞文" lang="sgs" hreflang="sgs" data-title="Luogarėtmos" data-language-autonym="Žemaitėška" data-language-local-name="薩莫吉希亞文" 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 – 白俄罗斯语" lang="be" hreflang="be" data-title="Лагарыфм" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" 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 – 保加利亚语" lang="bg" hreflang="bg" data-title="Логаритъм" data-language-autonym="Български" data-language-local-name="保加利亚语" 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 – 班亞爾文" lang="bjn" hreflang="bjn" data-title="Logaritma" data-language-autonym="Banjar" data-language-local-name="班亞爾文" 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 – 孟加拉语" lang="bn" hreflang="bn" data-title="লগারিদম" data-language-autonym="বাংলা" data-language-local-name="孟加拉语" 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 – 布列塔尼语" lang="br" hreflang="br" data-title="Logaritm" data-language-autonym="Brezhoneg" data-language-local-name="布列塔尼语" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Logaritam" title="Logaritam – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Logaritam" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B0%D1%80%D0%B8%D1%84%D0%BC" title="Логарифм – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Логарифм" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Logaritme" title="Logaritme – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Logaritme" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" 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 – 中库尔德语" lang="ckb" hreflang="ckb" data-title="لۆگاریتم" data-language-autonym="کوردی" data-language-local-name="中库尔德语" 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 – 捷克语" lang="cs" hreflang="cs" data-title="Logaritmus" data-language-autonym="Čeština" data-language-local-name="捷克语" 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 – 楚瓦什语" lang="cv" hreflang="cv" data-title="Логарифм" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" 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 – 威尔士语" lang="cy" hreflang="cy" data-title="Logarithm" data-language-autonym="Cymraeg" data-language-local-name="威尔士语" 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 – 丹麦语" lang="da" hreflang="da" data-title="Logaritme" data-language-autonym="Dansk" data-language-local-name="丹麦语" 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 – 德语" lang="de" hreflang="de" data-title="Logarithmus" data-language-autonym="Deutsch" data-language-local-name="德语" 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 – 希腊语" lang="el" hreflang="el" data-title="Λογάριθμος" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" 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="典范条目"><a href="https://en.wikipedia.org/wiki/Logarithm" title="Logarithm – 英语" lang="en" hreflang="en" data-title="Logarithm" data-language-autonym="English" data-language-local-name="英语" 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 – 世界语" lang="eo" hreflang="eo" data-title="Logaritmo" data-language-autonym="Esperanto" data-language-local-name="世界语" 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 – 西班牙语" lang="es" hreflang="es" data-title="Logaritmo" data-language-autonym="Español" data-language-local-name="西班牙语" 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 – 爱沙尼亚语" lang="et" hreflang="et" data-title="Logaritm" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" 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 – 巴斯克语" lang="eu" hreflang="eu" data-title="Logaritmo" data-language-autonym="Euskara" data-language-local-name="巴斯克语" 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 – 埃斯特雷馬杜拉文" lang="ext" hreflang="ext" data-title="Logaritmu" data-language-autonym="Estremeñu" data-language-local-name="埃斯特雷馬杜拉文" 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 – 波斯语" lang="fa" hreflang="fa" data-title="لگاریتم" data-language-autonym="فارسی" data-language-local-name="波斯语" 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 – 芬兰语" lang="fi" hreflang="fi" data-title="Logaritmi" data-language-autonym="Suomi" data-language-local-name="芬兰语" 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 – 法罗语" lang="fo" hreflang="fo" data-title="Logaritma" data-language-autonym="Føroyskt" data-language-local-name="法罗语" 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 – 法语" lang="fr" hreflang="fr" data-title="Logarithme" data-language-autonym="Français" data-language-local-name="法语" 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 – 爱尔兰语" lang="ga" hreflang="ga" data-title="Logartam" data-language-autonym="Gaeilge" data-language-local-name="爱尔兰语" 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 – 赣语" lang="gan" hreflang="gan" data-title="對數" data-language-autonym="贛語" data-language-local-name="赣语" 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 – 加利西亚语" lang="gl" hreflang="gl" data-title="Logaritmo" data-language-autonym="Galego" data-language-local-name="加利西亚语" 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 – 希伯来语" lang="he" hreflang="he" data-title="לוגריתם" data-language-autonym="עברית" data-language-local-name="希伯来语" 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 – 印地语" lang="hi" hreflang="hi" data-title="लघुगणक" data-language-autonym="हिन्दी" data-language-local-name="印地语" 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 – 斐濟印地文" lang="hif" hreflang="hif" data-title="Logarithm" data-language-autonym="Fiji Hindi" data-language-local-name="斐濟印地文" 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 – 克罗地亚语" lang="hr" hreflang="hr" data-title="Logaritam" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu badge-Q17437796 badge-featuredarticle mw-list-item" title="典范条目"><a href="https://hu.wikipedia.org/wiki/Logaritmus" title="Logaritmus – 匈牙利语" lang="hu" hreflang="hu" data-title="Logaritmus" data-language-autonym="Magyar" data-language-local-name="匈牙利语" 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 – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Լոգարիթմ" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" 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 – 国际语" lang="ia" hreflang="ia" data-title="Logarithmo" data-language-autonym="Interlingua" data-language-local-name="国际语" 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 – 印度尼西亚语" lang="id" hreflang="id" data-title="Logaritma" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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 – 伊多语" lang="io" hreflang="io" data-title="Logaritmo" data-language-autonym="Ido" data-language-local-name="伊多语" 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 – 冰岛语" lang="is" hreflang="is" data-title="Logri" data-language-autonym="Íslenska" data-language-local-name="冰岛语" 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 – 意大利语" lang="it" hreflang="it" data-title="Logaritmo" data-language-autonym="Italiano" data-language-local-name="意大利语" 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 – 日语" lang="ja" hreflang="ja" data-title="対数" data-language-autonym="日本語" data-language-local-name="日语" 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 – 牙買加克里奧爾英文" lang="jam" hreflang="jam" data-title="Lagaridim" data-language-autonym="Patois" data-language-local-name="牙買加克里奧爾英文" 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 – 格鲁吉亚语" lang="ka" hreflang="ka" data-title="ლოგარითმი" data-language-autonym="ქართული" data-language-local-name="格鲁吉亚语" 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 – 哈萨克语" lang="kk" hreflang="kk" data-title="Логарифм" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" 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) – 韩语" lang="ko" hreflang="ko" data-title="로그 (수학)" data-language-autonym="한국어" data-language-local-name="韩语" 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 – 拉丁语" lang="la" hreflang="la" data-title="Logarithmus" data-language-autonym="Latina" data-language-local-name="拉丁语" 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 – 新共同語言" lang="lfn" hreflang="lfn" data-title="Logaritmo" data-language-autonym="Lingua Franca Nova" data-language-local-name="新共同語言" 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 – 倫巴底文" lang="lmo" hreflang="lmo" data-title="Logaritm" data-language-autonym="Lombard" data-language-local-name="倫巴底文" 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 – 立陶宛语" lang="lt" hreflang="lt" data-title="Logaritmas" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" 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 – 拉脱维亚语" lang="lv" hreflang="lv" data-title="Logaritms" data-language-autonym="Latviešu" data-language-local-name="拉脱维亚语" 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 – 马拉加斯语" lang="mg" hreflang="mg" data-title="Anisa" data-language-autonym="Malagasy" data-language-local-name="马拉加斯语" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437796 badge-featuredarticle mw-list-item" title="典范条目"><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 – 马其顿语" lang="mk" hreflang="mk" data-title="Логаритам" data-language-autonym="Македонски" data-language-local-name="马其顿语" 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 – 马拉雅拉姆语" lang="ml" hreflang="ml" data-title="ലോഗരിതം" data-language-autonym="മലയാളം" data-language-local-name="马拉雅拉姆语" 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 – 马拉地语" lang="mr" hreflang="mr" data-title="लॉगॅरिदम" data-language-autonym="मराठी" data-language-local-name="马拉地语" 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 – 马来语" lang="ms" hreflang="ms" data-title="Logaritma" data-language-autonym="Bahasa Melayu" data-language-local-name="马来语" 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 – 缅甸语" lang="my" hreflang="my" data-title="လော့ဂရစ်သမ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="缅甸语" 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 – 低地德语" lang="nds" hreflang="nds" data-title="Logarithmus" data-language-autonym="Plattdüütsch" data-language-local-name="低地德语" 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 – 荷兰语" lang="nl" hreflang="nl" data-title="Logaritme" data-language-autonym="Nederlands" data-language-local-name="荷兰语" 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 – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Logaritme" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" 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 – 书面挪威语" lang="nb" hreflang="nb" data-title="Logaritme" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" 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 – 奥克语" lang="oc" hreflang="oc" data-title="Logaritme" data-language-autonym="Occitan" data-language-local-name="奥克语" 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 – 奥罗莫语" lang="om" hreflang="om" data-title="Loogarizimii" data-language-autonym="Oromoo" data-language-local-name="奥罗莫语" 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 – 旁遮普语" lang="pa" hreflang="pa" data-title="ਲਘੂਗਣਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="旁遮普语" 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 – 波兰语" lang="pl" hreflang="pl" data-title="Logarytm" data-language-autonym="Polski" data-language-local-name="波兰语" 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="典范条目"><a href="https://pt.wikipedia.org/wiki/Logaritmo" title="Logaritmo – 葡萄牙语" lang="pt" hreflang="pt" data-title="Logaritmo" data-language-autonym="Português" data-language-local-name="葡萄牙语" 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 – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Logaritm" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="典范条目"><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 – 俄语" lang="ru" hreflang="ru" data-title="Логарифм" data-language-autonym="Русский" data-language-local-name="俄语" 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 – 萨哈语" lang="sah" hreflang="sah" data-title="Логарифм" data-language-autonym="Саха тыла" data-language-local-name="萨哈语" 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 – 西西里语" lang="scn" hreflang="scn" data-title="Logaritmu" data-language-autonym="Sicilianu" data-language-local-name="西西里语" 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 – 塞尔维亚-克罗地亚语" lang="sh" hreflang="sh" data-title="Логаритам" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞尔维亚-克罗地亚语" 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 – 僧伽罗语" lang="si" hreflang="si" data-title="ලඝු ගණක" data-language-autonym="සිංහල" data-language-local-name="僧伽罗语" 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 – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Logaritmus" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" 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 – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Logaritem" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" 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 – 绍纳语" lang="sn" hreflang="sn" data-title="Daraunene" data-language-autonym="ChiShona" data-language-local-name="绍纳语" 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 – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Logaritmet" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" 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 – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Логаритам" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" 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 – 瑞典语" lang="sv" hreflang="sv" data-title="Logaritm" data-language-autonym="Svenska" data-language-local-name="瑞典语" 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 – 斯瓦希里语" lang="sw" hreflang="sw" data-title="Logi" data-language-autonym="Kiswahili" data-language-local-name="斯瓦希里语" 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 – 泰米尔语" lang="ta" hreflang="ta" data-title="மடக்கை" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" 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 – 泰语" lang="th" hreflang="th" data-title="ลอการิทึม" data-language-autonym="ไทย" data-language-local-name="泰语" 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 – 他加禄语" lang="tl" hreflang="tl" data-title="Logaritmo" data-language-autonym="Tagalog" data-language-local-name="他加禄语" 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 – 土耳其语" lang="tr" hreflang="tr" data-title="Logaritma" data-language-autonym="Türkçe" data-language-local-name="土耳其语" 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 – 鞑靼语" lang="tt" hreflang="tt" data-title="Логарифм" data-language-autonym="Татарча / tatarça" data-language-local-name="鞑靼语" 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 – 乌克兰语" lang="uk" hreflang="uk" data-title="Логарифм" data-language-autonym="Українська" data-language-local-name="乌克兰语" 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 – 乌尔都语" lang="ur" hreflang="ur" data-title="لاگرتھم" data-language-autonym="اردو" data-language-local-name="乌尔都语" 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 – 乌兹别克语" lang="uz" hreflang="uz" data-title="Logarifm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="典范条目"><a href="https://vi.wikipedia.org/wiki/Logarit" title="Logarit – 越南语" lang="vi" hreflang="vi" data-title="Logarit" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" 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 – 瓦瑞语" lang="war" hreflang="war" data-title="Logaritmo" data-language-autonym="Winaray" data-language-local-name="瓦瑞语" 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 – 吴语" lang="wuu" hreflang="wuu" data-title="对数" data-language-autonym="吴语" data-language-local-name="吴语" 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 – 意第绪语" lang="yi" hreflang="yi" data-title="לאגאריטם" data-language-autonym="ייִדיש" data-language-local-name="意第绪语" class="interlanguage-link-target"><span>ייִדיש</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 – 闽南语" lang="nan" hreflang="nan" data-title="Tùi-sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="闽南语" 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 – 粤语" lang="yue" hreflang="yue" data-title="對數" data-language-autonym="粵語" data-language-local-name="粤语" 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="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav 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(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ambox{border-left-color:#36c!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy,html.skin-theme-clientpref-os .mw-parser-output .ambox-delete{border-left-color:#b32424!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-speedy{background-color:#300!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-content{border-left-color:#f28500!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-style{border-left-color:#fc3!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-move{border-left-color:#9932cc!important}html.skin-theme-clientpref-os .mw-parser-output .ambox-protection{border-left-color:#a2a9b1!important}}</style><table class="box-问题条目 plainlinks metadata ambox ambox-content ambox-multiple_issues compact-ambox" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span"><div class="multiple-issues-text mw-collapsible"><b>本條目存在以下問題</b>，請協助<b><a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit">改善本條目</a></b>或在<b><a href="/wiki/Talk:%E5%AF%B9%E6%95%B0" title="Talk:对数">討論頁</a></b>針對議題發表看法。 <div class="mw-collapsible-content"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83732972"><table class="box-Refimprove plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Tango-nosources.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/45px-Tango-nosources.svg.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/68px-Tango-nosources.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Tango-nosources.svg/90px-Tango-nosources.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此條目<b>需要补充更多<a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源">来源</a></b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2019年9月17日</span>)</i></small><span class="hide-when-compact"><br /><small>请协助補充多方面<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">可靠来源</a>以<a class="external text" href="https://zh.wikipedia.org/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit">改善这篇条目</a>，<a href="/wiki/Wikipedia:%E5%8F%AF%E4%BE%9B%E6%9F%A5%E8%AF%81" class="mw-redirect" title="Wikipedia:可供查证">无法查证</a>的内容可能會因為<a href="/wiki/Template:Fact" class="mw-redirect" title="Template:Fact">异议提出</a>而被移除。<br />致使用者：请搜索一下条目的标题（来源搜索：<span class="plainlinks"><a rel="nofollow" class="external text" href="//www.google.com/search?&as_eq=wikipedia&q=%22%E5%AF%B9%E6%95%B0%22">"对数"</a> — <a rel="nofollow" class="external text" href="//www.google.com/search?q=%22%E5%AF%B9%E6%95%B0%22">网页</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=nws&q=&as_src=-newswire+-wire+-presswire+-PR+-press+-release+-wikipedia&q=%22%E5%AF%B9%E6%95%B0%22">新闻</a>、<a rel="nofollow" class="external text" href="//books.google.com/books?&as_brr=0&as_pub=-icon&q=%22%E5%AF%B9%E6%95%B0%22">书籍</a>、<a rel="nofollow" class="external text" href="//scholar.google.com/scholar?&q=%22%E5%AF%B9%E6%95%B0%22">学术</a>、<a rel="nofollow" class="external text" href="//www.google.com/search?tbm=isch&safe=off&q=%22%E5%AF%B9%E6%95%B0%22">图像</a></span>），以检查网络上是否存在该主题的更多可靠来源（<a href="/wiki/Wikipedia:%E5%8F%AF%E9%9D%A0%E6%9D%A5%E6%BA%90" title="Wikipedia:可靠来源">判定指引</a>）。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r83732972"><table class="box-Lead_too_short plainlinks metadata ambox ambox-content ambox-lead_too_short" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/40px-Wiki_letter_w.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/60px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Wiki_letter_w.svg/80px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">此条目<b><a href="/wiki/Wikipedia:%E6%A0%BC%E5%BC%8F%E6%89%8B%E5%86%8A/%E5%BA%8F%E8%A8%80%E7%AB%A0%E7%AF%80" title="Wikipedia:格式手冊/序言章節">序言章节</a>没有充分<a href="/wiki/Wikipedia:%E6%91%98%E8%A6%81%E6%A0%BC%E5%BC%8F" title="Wikipedia:摘要格式">总结</a>全文内容要点</b>。<span class="hide-when-compact"></span> <small class="date-container"><i>(<span class="date">2022年10月23日</span>)</i></small><span class="hide-when-compact"><br /><small>请考虑扩充序言，<a href="/wiki/Wikipedia:%E6%A0%BC%E5%BC%8F%E6%89%8B%E5%86%8A/%E5%BA%8F%E8%A8%80%E7%AB%A0%E7%AF%80#提供易讀的總覽" title="Wikipedia:格式手冊/序言章節">清晰概述</a>条目所有重點。请在条目的<a href="/wiki/Talk:%E5%AF%B9%E6%95%B0" title="Talk:对数">讨论页</a>讨论此问题。</small></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> </div></div><span class="hide-when-compact"></span><span class="hide-when-compact"></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Logarithm.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Logarithm.svg/300px-Logarithm.svg.png" decoding="async" width="300" height="360" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Logarithm.svg/450px-Logarithm.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Logarithm.svg/600px-Logarithm.svg.png 2x" data-file-width="542" data-file-height="650" /></a><figcaption>各种底数的对数函数图像：<span style="color: red">红色</span>函数底数是「<span style="color:red"><i>e</i></span>」, <span style="color:green">绿色</span>函数底数是<span style="color: green">2</span>，<span style="color:blue">蓝色</span>函数底数是<span style="color: blue">0.5</span>，刻度是半个单位。<span id="noteTag-cite_ref-sup"><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>註 1<span class="cite-bracket">]</span></a></sup></span></figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output 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\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{项 }}\,+\,{\text{项}}\\\scriptstyle {\text{加 数 }}\,+\,{\text{加 数}}\\\scriptstyle {\text{被 加 数 }}\,+\,{\text{加 数}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>项 </mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>项</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>加 数 </mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>加 数</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>被 加 数 </mtext> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>加 数</mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{项 }}\,+\,{\text{项}}\\\scriptstyle {\text{加数 }}\,+\,{\text{加数}}\\\scriptstyle {\text{被加数 }}\,+\,{\text{加数}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d361920fc9807d8e7c09ea9fb5af519e7df8a42e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.7ex; margin-bottom: -0.304ex; width:13.649ex; height:8.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{项 }}\,+\,{\text{项}}\\\scriptstyle {\text{加数 }}\,+\,{\text{加数}}\\\scriptstyle {\text{被加数 }}\,+\,{\text{加数}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;;"> <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 \scriptstyle {\text{和 }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>和 </mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{和 }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1b782d1169e35427cca1c598864830dbb1c5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.582ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\text{和 }}}"></span></td></tr><tr><th colspan="4" class="infobox-header" style=""><a href="/wiki/%E6%B8%9B%E6%B3%95" title="減法">減法</a> (−)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;;"> <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 \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{项 }}\,-\,{\text{项}}\\\scriptstyle {\text{被 减 数 }}\,-\,{\text{减 数 }}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>项 </mtext> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>项</mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>被 减 数 </mtext> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>减 数 </mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{项 }}\,-\,{\text{项}}\\\scriptstyle {\text{被减数 }}\,-\,{\text{减数 }}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1da4fe9f4ba71629e9a4006b9c562d2c4d0a37b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.093ex; height:5.676ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{项 }}\,-\,{\text{项}}\\\scriptstyle {\text{被减数 }}\,-\,{\text{减数 }}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;;"> <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 \scriptstyle {\text{差 }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>差 </mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{差 }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280f5f69d9ceb3ab159fa89745a40a635fd7f926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.582ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\text{差 }}}"></span></td></tr><tr><th colspan="4" class="infobox-header" style=""><a href="/wiki/%E4%B9%98%E6%B3%95" title="乘法">乘法</a> (×)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;;"> <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 \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{因 数 }}\,\times \,{\text{因 数 }}\\\scriptstyle {\text{被 乘 数 }}\,\times \,{\text{乘 数 }}\\\scriptstyle {\text{（ 英 文 中 ） }}{\text{乘 数 }}\,\times \,{\text{被 乘 数 }}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>因 数 </mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>因 数 </mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>被 乘 数 </mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>乘 数 </mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>（ 英 文 中 ） </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>乘 数 </mtext> </mrow> <mspace width="thinmathspace" /> <mo>×</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>被 乘 数 </mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{因数 }}\,\times \,{\text{因数 }}\\\scriptstyle {\text{被乘数 }}\,\times \,{\text{乘数 }}\\\scriptstyle {\text{（英文中） }}{\text{乘数 }}\,\times \,{\text{被乘数 }}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae06bdc935866bc4cd710f6279045b4a3f5904ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.7ex; margin-bottom: -0.304ex; width:22.142ex; height:8.843ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{因数 }}\,\times \,{\text{因数 }}\\\scriptstyle {\text{被乘数 }}\,\times \,{\text{乘数 }}\\\scriptstyle {\text{（英文中） }}{\text{乘数 }}\,\times \,{\text{被乘数 }}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;;"> <a href="/wiki/%E7%A7%AF" title="积"><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 \scriptstyle {\text{积 }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>积 </mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{积 }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5da1eab951b00183e07828f70b7895e3cc7bd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.582ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\text{积 }}}"></span></a></td></tr><tr><th colspan="4" class="infobox-header" style=""><a href="/wiki/%E9%99%A4%E6%B3%95" title="除法">除法</a> (÷)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;;"> <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 \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{被 除 数 }}}{\scriptstyle {\text{除 数 }}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{分 子 }}}{\scriptstyle {\text{分 母 }}}}\end{matrix}}\right\}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>被 除 数 </mtext> </mrow> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>除 数 </mtext> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>分 子 </mtext> </mrow> </mstyle> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>分 母 </mtext> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{被除数 }}}{\scriptstyle {\text{除数 }}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{分子 }}}{\scriptstyle {\text{分母 }}}}\end{matrix}}\right\}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcdf3d748da25a1322dc4f98c18577415e825ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:9.848ex; height:14.176ex;" alt="{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{被除数 }}}{\scriptstyle {\text{除数 }}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{分子 }}}{\scriptstyle {\text{分母 }}}}\end{matrix}}\right\}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;;"> <a href="/wiki/%E5%95%86%E6%95%B8" title="商數"><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 {\begin{matrix}\scriptstyle {\text{商 }}\\\scriptstyle {\text{比 值 }}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>商 </mtext> </mrow> </mstyle> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>比 值 </mtext> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\scriptstyle {\text{商 }}\\\scriptstyle {\text{比值 }}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6782bba4aacc19b46da5a3fc3ad1c952ef3ed600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:3.916ex; height:6.509ex;" alt="{\displaystyle {\begin{matrix}\scriptstyle {\text{商 }}\\\scriptstyle {\text{比值 }}\end{matrix}}}"></span></a></td></tr><tr><th colspan="4" class="infobox-header" style=""><a href="/wiki/%E6%A8%A1%E9%99%A4" title="模除">模除</a> (mod)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;;"> <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 \scriptstyle {\text{被 除 数 }}\,mod\,{\text{除 数}}\,\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>被 除 数 </mtext> </mrow> <mspace width="thinmathspace" /> <mi>m</mi> <mi>o</mi> <mi>d</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>除 数</mtext> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{被除数 }}\,mod\,{\text{除数}}\,\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e96263e7e4333debcce657f36071672297afaaf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.645ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\text{被除数 }}\,mod\,{\text{除数}}\,\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;;"> <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 \scriptstyle {\text{余 数 }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>余 数 </mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{余数 }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac86ca5f38cee7ddaa353bcae09e4cae90b004e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.165ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\text{余数 }}}"></span></td></tr><tr><th colspan="4" class="infobox-header" style=""><a href="/wiki/%E5%86%AA" title="冪">乘方</a> (^)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;;"> <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 \scriptstyle {\text{底 数 }}^{\text{指 数 }}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>底 数 </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>指 数 </mtext> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{底数 }}^{\text{指数 }}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/074053045b743a38f0b121e72947ae4047fa51b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.17ex; height:3.343ex;" alt="{\displaystyle \scriptstyle {\text{底数 }}^{\text{指数 }}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;;"> <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 \scriptstyle {\text{幂 }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>幂 </mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{幂 }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a68105faaac32cb2f4f0bf71a4af1de8d0a4392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.582ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\text{幂 }}}"></span></td></tr><tr><th colspan="4" class="infobox-header" style=""><a href="/wiki/%E6%96%B9%E6%A0%B9" title="方根"><i>n</i> 次方根</a> (√)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;;"> <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 \scriptstyle {\sqrt[{\text{根 指 数 }}]{\scriptstyle {\text{被 开 方 数 }}}}\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>被 开 方 数 </mtext> </mrow> </mstyle> <mrow class="MJX-TeXAtom-ORD"> <mtext>根 指 数 </mtext> </mrow> </mroot> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\sqrt[{\text{根指数 }}]{\scriptstyle {\text{被开方数 }}}}\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/129356d757065c14aa96f75e3c93198f2e4fba94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.558ex; height:3.676ex;" alt="{\displaystyle \scriptstyle {\sqrt[{\text{根指数 }}]{\scriptstyle {\text{被开方数 }}}}\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;;"> <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 \scriptstyle {\text{根 }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>根 </mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{根 }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ceafd8f3f39c0ee6232a0d300c34a1ba810a46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.582ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\text{根 }}}"></span></td></tr><tr><th colspan="4" class="infobox-header" style=""><a class="mw-selflink selflink">对数</a> (log)</th></tr><tr><th scope="row" class="infobox-label" style="display:none;;"></th><td class="infobox-data infobox-data-a" style="text-align:right; vertical-align:middle;;"> <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 \scriptstyle \log _{\text{底 }}({\text{真 数 }})\,=\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>底 </mtext> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>真 数 </mtext> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \log _{\text{底 }}({\text{真数 }})\,=\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ac3a27e34072224c2ecfae1c69fab033394c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.156ex; height:3.009ex;" alt="{\displaystyle \scriptstyle \log _{\text{底 }}({\text{真数 }})\,=\,}"></span></td><td class="infobox-data infobox-data-b" style="text-align:left; vertical-align:middle;;"> <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 \scriptstyle {\text{对 数 }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mtext>对 数 </mtext> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\text{对数 }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3181aa26d7fceaa0e8f3ac4d9a1c372e7e746ac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.165ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\text{对数 }}}"></span></td></tr></tbody></table></td> </tr><tr><td class="sidebar-navbar" style="line-height:1.6"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84244141"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E7%AE%97%E6%9C%AF%E8%BF%90%E7%AE%97" title="Template:算术运算"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:%E7%AE%97%E6%9C%AF%E8%BF%90%E7%AE%97" title="Template talk:算术运算"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%AE%97%E6%9C%AF%E8%BF%90%E7%AE%97" title="Special:编辑页面/Template:算术运算"><abbr title="编辑该模板">编</abbr></a></li></ul></div></td></tr></tbody></table> <p>在数学中，<b>對數</b>（英語：<span lang="en">logarithm</span>）是<a href="/wiki/%E5%86%AA" title="冪">冪運算</a>的逆運算。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="定义"><span id=".E5.AE.9A.E4.B9.89"></span>定义</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=1" title="编辑章节：定义"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>当<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=\beta ^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\beta ^{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dd9928e3008e75357d97b12c82cb90b38fdd137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.814ex; height:2.676ex;" alt="{\displaystyle x=\beta ^{y}}"></span>时，則有 </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 y=\log _{\beta }x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\log _{\beta }x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c992a796fab2cfe00f1f610cbfe681d23fd04e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-right: -0.271ex; width:10ex; height:3.009ex;" alt="{\displaystyle y=\log _{\beta }x\!}"></span></dd></dl> <p>其中<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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>是對數的<a href="/wiki/%E5%BA%95%E6%95%B8_(%E6%8C%87%E6%95%B8)" title="底數 (指數)">底</a>（也稱為基數），而 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>就是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>（对于底数<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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>）的对数，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>也称为<a href="/wiki/%E7%9C%9F%E6%95%B0" class="mw-redirect" title="真数">真数</a>。 </p><p>底数<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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>的值在实数范围内常取<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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>、 10、2等，但一定不能是1或0<span id="noteTag-cite_ref-sup"><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>註 2<span class="cite-bracket">]</span></a></sup></span> </p><p>当<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></i>和<i><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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span></i>进一步限制为正<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a>的时候，对数是唯一的实数。例如，因为 </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 3^{4}=3\times 3\times 3\times 3=81}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>×</mo> <mn>3</mn> <mo>=</mo> <mn>81</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{4}=3\times 3\times 3\times 3=81}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a6e4bf3de42f59ea0da368bcb0441aea133d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.91ex; height:2.676ex;" alt="{\displaystyle 3^{4}=3\times 3\times 3\times 3=81}"></span>，</dd></dl> <p>我们可以得出 </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 4=\log _{3}81\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⁡</mo> <mn>81</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4=\log _{3}81\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3904f8093cb666f092b5a57ac28b0c42b3caed99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.218ex; width:10.829ex; height:2.676ex;" alt="{\displaystyle 4=\log _{3}81\!}"></span>，</dd></dl> <p>用日常语言说，即「81以3为底的对数是4」。这个意思就是说，3的4次方是81。 </p> <div class="mw-heading mw-heading2"><h2 id="历史"><span id=".E5.8E.86.E5.8F.B2"></span>历史</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=2" title="编辑章节：历史"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="对数"><span id=".E5.AF.B9.E6.95.B0"></span>对数</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=3" title="编辑章节：对数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>15世纪时，法国数学家<span class="ilh-all" data-orig-title="尼古拉·丘凯" data-lang-code="en" data-lang-name="英语" data-foreign-title="Nicolas Chuquet"><span class="ilh-page"><a href="/w/index.php?title=%E5%B0%BC%E5%8F%A4%E6%8B%89%C2%B7%E4%B8%98%E5%87%AF&action=edit&redlink=1" class="new" title="尼古拉·丘凯（页面不存在）">尼古拉·丘凯</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Nicolas_Chuquet" class="extiw" title="en:Nicolas Chuquet"><span lang="en" dir="auto">Nicolas Chuquet</span></a></span>）</span></span>和德国数学家<span class="ilh-all" data-orig-title="米夏埃尔·施蒂费尔" data-lang-code="en" data-lang-name="英语" data-foreign-title="Michael Stifel"><span class="ilh-page"><a href="/w/index.php?title=%E7%B1%B3%E5%A4%8F%E5%9F%83%E5%B0%94%C2%B7%E6%96%BD%E8%92%82%E8%B4%B9%E5%B0%94&action=edit&redlink=1" class="new" title="米夏埃尔·施蒂费尔（页面不存在）">米夏埃尔·施蒂费尔</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Michael_Stifel" class="extiw" title="en:Michael Stifel"><span lang="en" dir="auto">Michael Stifel</span></a></span>）</span></span>在开展研究工作时产生了发展对数的思想，他们，尤其是后者，对等差数列和<a href="/wiki/%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97" title="等比数列">等比数列</a>的关系作了一些研究。但他们并没有使其得到更进一步的发展。<sup id="cite_ref-上海交通大学数学科学学院_3-0" class="reference"><a href="#cite_note-上海交通大学数学科学学院-3"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>一般认为对数于16世纪末至17世纪初期间由苏格兰数学家<a href="/wiki/%E7%B4%84%E7%BF%B0%C2%B7%E7%B4%8D%E7%9A%AE%E7%88%BE" title="約翰·納皮爾">约翰·纳皮尔</a>男爵和瑞士工程师<a href="/wiki/%E7%BA%A6%E6%96%AF%E7%89%B9%C2%B7%E6%AF%94%E5%B0%94%E5%90%89" title="约斯特·比尔吉">约斯特·比尔吉</a>发明。比尔吉曾担任过著名天文学家<a href="/wiki/%E5%BC%80%E6%99%AE%E5%8B%92" class="mw-redirect" title="开普勒">开普勒</a>的助手，因此会经常接触到复杂的天文计算，他也因此产生了化简数值计算的想法。<span id="noteTag-cite_ref-sup"><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>註 3<span class="cite-bracket">]</span></a></sup></span>纳皮尔是一位苏格兰贵族，对数值的计算有很深的研究。为了找到简化<a href="/wiki/%E7%90%83%E9%9D%A2%E4%B8%89%E8%A7%92" class="mw-redirect" title="球面三角">球面三角</a>计算的方法，他也产生了发展对数的想法。1614年，他在自己的书籍《奇妙的对数表的描述》<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>上发布了自己的对数表，相較比尔吉早了6年。纳皮尔发明的<a href="/wiki/%E7%BA%B3%E7%9A%AE%E5%B0%94%E7%AE%97%E7%AD%B9" class="mw-redirect" title="纳皮尔算筹">纳皮尔算筹</a>用加减法代替了乘除法，成功简化了乘除法的运算，他的对数被后人称为纳皮尔对数，记法为Nap·logx。<sup id="cite_ref-上海交通大学数学科学学院_3-1" class="reference"><a href="#cite_note-上海交通大学数学科学学院-3"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>1624年，英国数学家<span class="ilh-all" data-orig-title="亨利·布里格斯" data-lang-code="en" data-lang-name="英语" data-foreign-title="Henry Briggs (mathematician)"><span class="ilh-page"><a href="/w/index.php?title=%E4%BA%A8%E5%88%A9%C2%B7%E5%B8%83%E9%87%8C%E6%A0%BC%E6%96%AF&action=edit&redlink=1" class="new" title="亨利·布里格斯（页面不存在）">亨利·布里格斯</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Henry_Briggs_(mathematician)" class="extiw" title="en:Henry Briggs (mathematician)"><span lang="en" dir="auto">Henry Briggs (mathematician)</span></a></span>）</span></span>书籍《对数算术》成功出版，书中写有14位常用对数表。布里格斯率先采用了以10为底的<a href="/wiki/%E5%B8%B8%E7%94%A8%E5%B0%8D%E6%95%B8" title="常用對數">常用对数</a>，而现在它已通用。他还制作了正弦和正切的对数表。荷兰数学家兼出版商在布里格斯的基础上加以改进，他出版的数个对数表在欧洲迅速普及起来。<sup id="cite_ref-上海交通大学数学科学学院_3-2" class="reference"><a href="#cite_note-上海交通大学数学科学学院-3"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>17世纪中叶（清朝初年），中国数学家<a href="/wiki/%E8%96%9B%E9%B3%B3%E7%A5%9A" title="薛鳳祚">薛凤祚</a>和波兰传教士<a href="/wiki/%E7%A9%86%E5%B0%BC%E9%96%A3" title="穆尼閣">穆尼阁</a>合作完成了中国最早的对数著作《比例对数表》（又名《历学会通》），对数自此传入中国。<sup id="cite_ref-上海交通大学数学科学学院_3-3" class="reference"><a href="#cite_note-上海交通大学数学科学学院-3"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>此书称真数为“原数”，对数为“比例数”。而《数理精蕴》中则称作对数比例：“对数比例乃西士若往·纳白尔所作，以借数与真数对列成表，故名对数表。”中国因此普遍称之为“对数”。 </p><p>对数对科学的进步有所贡献，特别是对<a href="/wiki/%E5%A4%A9%E6%96%87%E5%AD%A6" class="mw-redirect" title="天文学">天文学</a>，使某些繁难的乘法计算转换为加法计算。在计算器和计算机发明之前，对数长期用于测量、航海、和其他<a href="/wiki/%E5%BA%94%E7%94%A8%E6%95%B0%E5%AD%A6" title="应用数学">应用数学</a>分支中。 </p> <div class="mw-heading mw-heading3"><h3 id="符号"><span id=".E7.AC.A6.E5.8F.B7"></span>符号</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=4" title="编辑章节：符号"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>对数符号<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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4debd0ab1c6ce342d0172a7643733305c37bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }"></span>出自拉丁文logarithmus，最早由1632年<a href="/wiki/%E6%84%8F%E5%A4%A7%E5%88%A9" title="意大利">意大利</a>数学家<a href="/wiki/%E5%8D%A1%E7%93%A6%E5%88%97%E9%87%8C" class="mw-redirect" title="卡瓦列里">卡瓦列里</a>所使用。纳皮尔在表示对数时套用logarithm整个词，并未作简化。1624年，<a href="/wiki/%E5%BC%80%E6%99%AE%E5%8B%92" class="mw-redirect" title="开普勒">开普勒</a>才把对数符号简化为<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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4debd0ab1c6ce342d0172a7643733305c37bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle \log }"></span>，<a href="/wiki/%E5%A8%81%E5%BB%89%C2%B7%E5%A5%A5%E7%89%B9%E9%9B%B7%E5%BE%B7" title="威廉·奥特雷德">奥特雷德</a>在1647年也用简化了的Log。 </p><p>1893年，<a href="/wiki/%E6%9C%B1%E5%A1%9E%E4%BD%A9%C2%B7%E7%9A%AE%E4%BA%9E%E8%AB%BE" title="朱塞佩·皮亞諾">皮亚诺</a>用<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed172b0f5195382a3500c713941f945ad4db3898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \ln x}"></span>及<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 \lg x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>lg</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lg x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d7d8f8fbd44c0f2edcc2c28e9e23f3b6fcc68a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.526ex; height:2.509ex;" alt="{\displaystyle \lg x}"></span>分别表示以<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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>为底的对数和以10为底的对数。1902年，<span class="ilh-all" data-orig-title="奥托·施托尔茨" data-lang-code="en" data-lang-name="英语" data-foreign-title="Otto Stolz"><span class="ilh-page"><a href="/w/index.php?title=%E5%A5%A5%E6%89%98%C2%B7%E6%96%BD%E6%89%98%E5%B0%94%E8%8C%A8&action=edit&redlink=1" class="new" title="奥托·施托尔茨（页面不存在）">施托尔茨</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Otto_Stolz" class="extiw" title="en:Otto Stolz"><span lang="en" dir="auto">Otto Stolz</span></a></span>）</span></span>等人以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\log .b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>log</mi> <mo>.</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\log .b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5fef772be92ba876c25e0ff44f0f9d9f9bd251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.62ex; height:2.509ex;" alt="{\displaystyle a\log .b}"></span>表示以<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>为底的<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>的对数。 </p><p>20世纪初，形成了对数的现代标准表示<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 _{\alpha }\mathrm {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }\mathrm {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea394670db41a4fe41616ce69c6c939fa7e5ca60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.386ex; height:2.676ex;" alt="{\displaystyle \log _{\alpha }\mathrm {N} }"></span>，为了使用方便，自然对数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a2c6c380596b8d0e34cafc070f8efdfa7b21559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.39ex; height:2.176ex;" alt="{\displaystyle \ln N}"></span>的记法得到了普遍认可。 </p> <div class="mw-heading mw-heading2"><h2 id="对数函数"><span id=".E5.AF.B9.E6.95.B0.E5.87.BD.E6.95.B0"></span>对数函数</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=5" title="编辑章节：对数函数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%AF%B9%E6%95%B0%E5%87%BD%E6%95%B0" class="mw-redirect" title="对数函数">对数函数</a></div> <p>函数<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 _{\alpha }x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73cba1bb23e2ad299695165fe67bdc9927b55db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.973ex; height:2.676ex;" alt="{\displaystyle \log _{\alpha }x}"></span>依赖于<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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>二者，但是术语<b>对数函数</b>在标准用法中用来称呼形如<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 _{\alpha }x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73cba1bb23e2ad299695165fe67bdc9927b55db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.973ex; height:2.676ex;" alt="{\displaystyle \log _{\alpha }x}"></span>的函数，在其中<b>底数</b><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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>是固定的而只有一个<a href="/wiki/%E5%8F%83%E6%95%B8_(%E6%95%B8%E5%AD%B8)" title="參數 (數學)">參數</a><i><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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span></i>。<span id="noteTag-cite_ref-sup"><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>註 4<span class="cite-bracket">]</span></a></sup></span> </p><p>对数函数图像和指数函数图像关于直线<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=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe2e7da593fb7b41d44e87a97fefdd8998b77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle y=x}"></span>对称，互为<a href="/wiki/%E9%80%86%E5%87%BD%E6%95%B0" class="mw-redirect" title="逆函数">逆函数</a>。 </p><p>对数函数的性质有： </p> <ol><li>都过<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 (1,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b53cc1773694affcc1d4d6c2c778d43156a1206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,0)}"></span>点；</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>即y軸為其垂直漸近線。</li> <li><a href="/wiki/%E5%AE%9A%E4%B9%89%E5%9F%9F" title="定义域">定义域</a>为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de77e40eb7e2582eef8a5a1da1bc027b7d9a8d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.138ex; height:2.843ex;" alt="{\displaystyle (0,+\infty )}"></span>，<a href="/wiki/%E5%80%BC%E5%9F%9F" title="值域">值域</a>为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>；</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d81dbbc4786493c7b8548cc324a978d7cf5dbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >1}"></span>，在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de77e40eb7e2582eef8a5a1da1bc027b7d9a8d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.138ex; height:2.843ex;" alt="{\displaystyle (0,+\infty )}"></span>上是增函数；<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 1>\alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>></mo> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1>\alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ace1795a5c2cf5c31afa61dbb753c5f01106e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.009ex; height:2.176ex;" alt="{\displaystyle 1>\alpha >0}"></span>时，在<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,+\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,+\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de77e40eb7e2582eef8a5a1da1bc027b7d9a8d6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.138ex; height:2.843ex;" alt="{\displaystyle (0,+\infty )}"></span>上是减函数。</li> <li>當 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<\alpha <e^{-e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<\alpha <e^{-e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2bd506d6c6ebf7dd4f1cf4924cca864a4a6189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.207ex; height:2.509ex;" alt="{\displaystyle 0<\alpha <e^{-e}}"></span>時和<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=\alpha ^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\alpha ^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797f11565df51770a27304ee77c1e66ca6974949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.914ex; height:2.676ex;" alt="{\displaystyle y=\alpha ^{x}}"></span>交於三點；<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 e^{-e}<\alpha <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>e</mi> </mrow> </msup> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-e}<\alpha <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad10ce3e416f9e275e7e91212ef5556940caead3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.207ex; height:2.509ex;" alt="{\displaystyle e^{-e}<\alpha <1}"></span>時交於一點；<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 1<\alpha <e^{\frac {1}{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1<\alpha <e^{\frac {1}{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0206b7b2d9e12769807a74879ab0d2c71fff408" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.666ex; height:3.343ex;" alt="{\displaystyle 1<\alpha <e^{\frac {1}{e}}}"></span>時交於兩點；<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 \alpha =e^{\frac {1}{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =e^{\frac {1}{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e1665858432af6bf1ac1497d73babaf73b643c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.405ex; height:3.343ex;" alt="{\displaystyle \alpha =e^{\frac {1}{e}}}"></span>時交於一點；<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 \alpha >e^{\frac {1}{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>e</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >e^{\frac {1}{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/495285daa9d96135b432030bf8fc5dd13f678982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.405ex; height:3.343ex;" alt="{\displaystyle \alpha >e^{\frac {1}{e}}}"></span>時則無交點。</li></ol> <div class="mw-heading mw-heading2"><h2 id="运算公式"><span id=".E8.BF.90.E7.AE.97.E5.85.AC.E5.BC.8F"></span>运算公式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=6" title="编辑章节：运算公式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>名稱 </th> <th>公式 </th> <th><a href="/wiki/%E6%95%B8%E5%AD%B8%E8%AD%89%E6%98%8E" title="數學證明">證明</a> </th></tr> <tr> <td>和差 </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 _{\alpha }MN=\log _{\alpha }\!M+\log _{\alpha }\!N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mi>M</mi> <mi>N</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>M</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }MN=\log _{\alpha }\!M+\log _{\alpha }\!N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01abe1c5eaac70d18fb24fd36deb3ed73bffcac4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.105ex; height:2.676ex;" alt="{\displaystyle \log _{\alpha }MN=\log _{\alpha }\!M+\log _{\alpha }\!N}"></span> </td> <td>設<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 M=\beta ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\beta ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14aca1c1c357feac8da9be79b81fa12924ccc87d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.553ex; height:2.676ex;" alt="{\displaystyle M=\beta ^{m}}"></span>，<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=\beta ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\beta ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe24c8c80a0866d4e52d3875e27fc31cf5bab1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.718ex; height:2.676ex;" alt="{\displaystyle N=\beta ^{n}}"></span> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\log _{\alpha }\ M\!N&=\log _{\alpha }\ \beta ^{m}\!\beta ^{n}\\&=\log _{\alpha }\ \beta ^{m+n}\\&=(m+n)\log _{\alpha }\!\beta \\&=m\log _{\alpha }\!\beta +n\log _{\alpha }\!\beta \\&=\log _{\alpha }\ \beta ^{m}+\log _{\alpha }\ \beta ^{n}\\&=\log _{\alpha }\!M+\log _{\alpha }\!N\\\log _{\alpha }\!{\frac {M}{N}}&=\log _{\alpha }\!M+\log _{\alpha }\!{\frac {1}{N}}\\&=\log _{\alpha }\!M-\log _{\alpha }\!N\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <mi>M</mi> <mspace width="negativethinmathspace" /> <mi>N</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>β</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>m</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>β</mi> <mo>+</mo> <mi>n</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>β</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>M</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>N</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mi>N</mi> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>M</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>M</mi> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>N</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log _{\alpha }\ M\!N&=\log _{\alpha }\ \beta ^{m}\!\beta ^{n}\\&=\log _{\alpha }\ \beta ^{m+n}\\&=(m+n)\log _{\alpha }\!\beta \\&=m\log _{\alpha }\!\beta +n\log _{\alpha }\!\beta \\&=\log _{\alpha }\ \beta ^{m}+\log _{\alpha }\ \beta ^{n}\\&=\log _{\alpha }\!M+\log _{\alpha }\!N\\\log _{\alpha }\!{\frac {M}{N}}&=\log _{\alpha }\!M+\log _{\alpha }\!{\frac {1}{N}}\\&=\log _{\alpha }\!M-\log _{\alpha }\!N\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f493f4da7c03b78989560e2852b5c56e487d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.005ex; width:32.048ex; height:27.176ex;" alt="{\displaystyle {\begin{aligned}\log _{\alpha }\ M\!N&=\log _{\alpha }\ \beta ^{m}\!\beta ^{n}\\&=\log _{\alpha }\ \beta ^{m+n}\\&=(m+n)\log _{\alpha }\!\beta \\&=m\log _{\alpha }\!\beta +n\log _{\alpha }\!\beta \\&=\log _{\alpha }\ \beta ^{m}+\log _{\alpha }\ \beta ^{n}\\&=\log _{\alpha }\!M+\log _{\alpha }\!N\\\log _{\alpha }\!{\frac {M}{N}}&=\log _{\alpha }\!M+\log _{\alpha }\!{\frac {1}{N}}\\&=\log _{\alpha }\!M-\log _{\alpha }\!N\end{aligned}}}"></span> </p> </td></tr> <tr> <td>基变換（换底公式） </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 _{\alpha }\!x={\frac {\log _{\beta }\!x}{\log _{\beta }\!\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>x</mi> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }\!x={\frac {\log _{\beta }\!x}{\log _{\beta }\!\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b53b8aba678315c2746a6f4ba088e94a312a52a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.154ex; height:6.843ex;" alt="{\displaystyle \log _{\alpha }\!x={\frac {\log _{\beta }\!x}{\log _{\beta }\!\alpha }}}"></span> </td> <td> <dl><dt>设 <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 _{\alpha }\!x=t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>x</mi> <mo>=</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }\!x=t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5afe983200fa978dd9a63376b3b375973f3058ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.524ex; height:2.676ex;" alt="{\displaystyle \log _{\alpha }\!x=t}"></span></dt> <dt>∴<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=\alpha ^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\alpha ^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72d4ab9389f3256da12b2d6e35710c4ef6025478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.742ex; height:2.509ex;" alt="{\displaystyle x=\alpha ^{t}}"></span></dt> <dt><b>对其两边取对数，则有</b><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 _{\beta }x=\log _{\beta }\alpha ^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\beta }x=\log _{\beta }\alpha ^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b98e5440dfed29b60ffcc8f886b6d220b34958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.808ex; height:3.343ex;" alt="{\displaystyle \log _{\beta }x=\log _{\beta }\alpha ^{t}}"></span></dt> <dt>即 <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 _{\beta }x=t\log _{\beta }\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>t</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\beta }x=t\log _{\beta }\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e12a4f3c029d87d508432cbc5976e0a197183282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.208ex; height:3.009ex;" alt="{\displaystyle \log _{\beta }x=t\log _{\beta }\alpha }"></span></dt> <dt>又∵ <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 _{\alpha }x=t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }x=t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e32414ee55c64b38a0f1892fc76371a1c86408" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.911ex; height:2.676ex;" alt="{\displaystyle \log _{\alpha }x=t}"></span></dt> <dt>∴ <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 _{\alpha }x={\frac {\log _{\beta }\!x}{\log _{\beta }\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>x</mi> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }x={\frac {\log _{\beta }\!x}{\log _{\beta }\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45bd57048435048bb8e8e8de352ca79df76541cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.928ex; height:6.843ex;" alt="{\displaystyle \log _{\alpha }x={\frac {\log _{\beta }\!x}{\log _{\beta }\alpha }}}"></span></dt></dl> </td></tr> <tr> <td>指係(次方公式) </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 _{\alpha ^{n}}x^{m}={\frac {m}{n}}\log _{\alpha }\!x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mo>⁡</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha ^{n}}x^{m}={\frac {m}{n}}\log _{\alpha }\!x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b5f45bc198b1674e0a7314d15092606d462c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.56ex; height:4.676ex;" alt="{\displaystyle \log _{\alpha ^{n}}x^{m}={\frac {m}{n}}\log _{\alpha }\!x}"></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 {\begin{aligned}\log _{\alpha ^{n}}\ {x^{m}}&={\frac {\ln \ x^{m}}{\ln \ \alpha ^{n}}}\\&={\frac {m\ln \!x}{n\ln \!\alpha }}\\&={\frac {m}{n}}\log _{\alpha }\!x\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mo>⁡</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mtext> </mtext> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mtext> </mtext> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>α</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>x</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log _{\alpha ^{n}}\ {x^{m}}&={\frac {\ln \ x^{m}}{\ln \ \alpha ^{n}}}\\&={\frac {m\ln \!x}{n\ln \!\alpha }}\\&={\frac {m}{n}}\log _{\alpha }\!x\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66f5e2e26131e3c86c3351101e1b1eadedbc692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.105ex; margin-bottom: -0.233ex; width:21.892ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}\log _{\alpha ^{n}}\ {x^{m}}&={\frac {\ln \ x^{m}}{\ln \ \alpha ^{n}}}\\&={\frac {m\ln \!x}{n\ln \!\alpha }}\\&={\frac {m}{n}}\log _{\alpha }\!x\end{aligned}}}"></span> </td></tr> <tr> <td>还原 </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 {\begin{aligned}\alpha ^{\log _{\alpha }\!x}&=\log _{\alpha }\!\alpha ^{x}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha ^{\log _{\alpha }\!x}&=\log _{\alpha }\!\alpha ^{x}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366654f4ea29ad202dd910bb8d4d9f0a1bc610e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.546ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}\alpha ^{\log _{\alpha }\!x}&=\log _{\alpha }\!\alpha ^{x}\end{aligned}}}"></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 {\begin{aligned}\alpha ^{\log _{\alpha }\!x}&=x\\&=\log _{\alpha }\!\alpha ^{x}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha ^{\log _{\alpha }\!x}&=x\\&=\log _{\alpha }\!\alpha ^{x}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e338393f6da09dc34467efa6fad5189087bbbba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.546ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\alpha ^{\log _{\alpha }\!x}&=x\\&=\log _{\alpha }\!\alpha ^{x}\end{aligned}}}"></span> </td></tr> <tr> <td>互換 </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 M^{\log _{\alpha }\!N}=N^{\log _{\alpha }\!M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>N</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>M</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{\log _{\alpha }\!N}=N^{\log _{\alpha }\!M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12af989d8ce20bf6c8eed40657a5352b7aff9f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.61ex; height:2.676ex;" alt="{\displaystyle M^{\log _{\alpha }\!N}=N^{\log _{\alpha }\!M}}"></span> </td> <td> <dl><dt>设 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\log _{\alpha }\!N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\log _{\alpha }\!N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f60ac907005f9483e61cb8b515a181f80bc92be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.416ex; height:2.676ex;" alt="{\displaystyle b={\log _{\alpha }\!N}}"></span>,<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\log _{\alpha }\!M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\log _{\alpha }\!M}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f0aa9a342db1e4ccf4a10e305d02fe224f61a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.803ex; height:2.676ex;" alt="{\displaystyle c={\log _{\alpha }\!M}}"></span></dt> <dt>则有 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=\alpha ^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\alpha ^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da36c278d34777e7ca939f42ecbcc2bc42cf0582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.587ex; height:2.676ex;" alt="{\displaystyle N=\alpha ^{b}}"></span>,<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\alpha ^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\alpha ^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700627947589e9e0d824ceb5cb46fcd7e20394b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.972ex; height:2.343ex;" alt="{\displaystyle M=\alpha ^{c}}"></span></dt> <dt>即 <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 M^{\log _{\alpha }\!N}=(\alpha ^{c})^{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>N</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{\log _{\alpha }\!N}=(\alpha ^{c})^{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43dbaaf8db89c4fd58cb28a1e0a5164963953c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.587ex; height:3.176ex;" alt="{\displaystyle M^{\log _{\alpha }\!N}=(\alpha ^{c})^{b}}"></span>,<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{\log _{\alpha }\!M}=(\alpha ^{b})^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>M</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N^{\log _{\alpha }\!M}=(\alpha ^{b})^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/609f70c41eeb09f003e75527567736d4a950a5a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.479ex; height:3.176ex;" alt="{\displaystyle N^{\log _{\alpha }\!M}=(\alpha ^{b})^{c}}"></span></dt></dl> </td></tr> <tr> <td>倒数 </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 _{\alpha }\!\theta ={\frac {1}{\log _{\theta }\!\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }\!\theta ={\frac {1}{\log _{\theta }\!\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a966c2235fb1d00c4ec0b10f45a08693cd0d780" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.744ex; height:5.843ex;" alt="{\displaystyle \log _{\alpha }\!\theta ={\frac {1}{\log _{\theta }\!\alpha }}}"></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 {\begin{aligned}\log _{\alpha }\!\theta &={\dfrac {1}{\dfrac {\ln \!\alpha }{\ln \!\theta }}}\\&={\frac {1}{\log _{\theta }\!\alpha }}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>θ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>α</mi> </mrow> <mrow> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>θ</mi> </mrow> </mfrac> </mstyle> </mfrac> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>α</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log _{\alpha }\!\theta &={\dfrac {1}{\dfrac {\ln \!\alpha }{\ln \!\theta }}}\\&={\frac {1}{\log _{\theta }\!\alpha }}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae9da27395fc63abdc892b36126f92349ff5e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.481ex; margin-bottom: -0.19ex; width:15.495ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\log _{\alpha }\!\theta &={\dfrac {1}{\dfrac {\ln \!\alpha }{\ln \!\theta }}}\\&={\frac {1}{\log _{\theta }\!\alpha }}\end{aligned}}}"></span> </td></tr> <tr> <td>链式 </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 {\begin{aligned}\log _{\gamma }\!\beta \log _{\beta }\!\alpha &=\log _{\gamma }\!\alpha \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>β</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>α</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>α</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log _{\gamma }\!\beta \log _{\beta }\!\alpha &=\log _{\gamma }\!\alpha \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86758fa3491a92017580bb1d4baacc692a038fd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.883ex; height:3.176ex;" alt="{\displaystyle {\begin{aligned}\log _{\gamma }\!\beta \log _{\beta }\!\alpha &=\log _{\gamma }\!\alpha \end{aligned}}}"></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 {\begin{aligned}\log _{\gamma }\!\beta \log _{\beta }\!\alpha &={\frac {\ln \!\alpha }{\ln \!\beta }}\ {\frac {\ln \!\beta }{\ln \!\gamma }}\\&={\frac {\ln \!\alpha }{\ln \!\gamma }}\\&=\log _{\gamma }\!\alpha \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>β</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>α</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>α</mi> </mrow> <mrow> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>β</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>β</mi> </mrow> <mrow> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>γ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>α</mi> </mrow> <mrow> <mi>ln</mi> <mspace width="negativethinmathspace" /> <mi>γ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <mi>α</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\log _{\gamma }\!\beta \log _{\beta }\!\alpha &={\frac {\ln \!\alpha }{\ln \!\beta }}\ {\frac {\ln \!\beta }{\ln \!\gamma }}\\&={\frac {\ln \!\alpha }{\ln \!\gamma }}\\&=\log _{\gamma }\!\alpha \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f94cd7af7da5d2514692ccc7a20eaaa10a3c919" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:24.25ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\log _{\gamma }\!\beta \log _{\beta }\!\alpha &={\frac {\ln \!\alpha }{\ln \!\beta }}\ {\frac {\ln \!\beta }{\ln \!\gamma }}\\&={\frac {\ln \!\alpha }{\ln \!\gamma }}\\&=\log _{\gamma }\!\alpha \end{aligned}}}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="有理和无理指数"><span id=".E6.9C.89.E7.90.86.E5.92.8C.E6.97.A0.E7.90.86.E6.8C.87.E6.95.B0"></span>有理和无理<a href="/wiki/%E6%8C%87%E6%95%B0%E5%87%BD%E6%95%B0" title="指数函数">指数</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=7" title="编辑章节：有理和无理指数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>如果<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>是<a href="/wiki/%E8%87%AA%E7%84%B6%E6%95%B8" class="mw-redirect" title="自然數">自然數</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 {\beta }^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\beta }^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111ce645005886962078c1c52a1124b5cc58bd6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.555ex; height:2.676ex;" alt="{\displaystyle {\beta }^{n}}"></span>表示等于<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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>个因子的<a href="/wiki/%E4%B9%98%E6%B3%95" title="乘法">乘积</a>： </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 {\beta }^{n}=\underbrace {\beta \times \beta \times \cdots \times \beta } _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>β</mi> <mo>×</mo> <mi>β</mi> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <mi>β</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\beta }^{n}=\underbrace {\beta \times \beta \times \cdots \times \beta } _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0abe63b94928984214a2ddae678ae7e2fb664f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-right: -0.017ex; width:20.912ex; height:6.009ex;" alt="{\displaystyle {\beta }^{n}=\underbrace {\beta \times \beta \times \cdots \times \beta } _{n}}"></span>。</dd></dl> <p>但是，如果<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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>是不等于1的正实数，这个定义可以扩展到在一个<a href="/wiki/%E5%9F%9F_(%E6%95%B0%E5%AD%A6)" title="域 (数学)">域</a>中的任何实数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>（参见<a href="/wiki/%E5%B9%82" class="mw-redirect" title="幂">幂</a>）。类似的，对数函数可以定义于任何正实数。对于不等于1的每个正底数<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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>，有一个对数<a href="/wiki/%E5%87%BD%E6%95%B0" title="函数">函数</a>和一个<a href="/wiki/%E6%8C%87%E6%95%B0%E5%87%BD%E6%95%B0" title="指数函数">指数函数</a>，它们互为<a href="/wiki/%E5%8F%8D%E5%87%BD%E6%95%B8" title="反函數">反函数</a>。 </p><p>对数可以简化乘法运算为加法，除法为减法，幂运算为乘法，根运算为除法。所以，在发明<a href="/wiki/%E7%94%B5%E5%AD%90%E8%AE%A1%E7%AE%97%E6%9C%BA" title="电子计算机">电子计算机</a>之前，对数对进行冗长的数值运算是很有用的，它们广泛的用于<a href="/wiki/%E5%A4%A9%E6%96%87" class="mw-redirect" title="天文">天文</a>、<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%A6" title="工程学">工程</a>、<a href="/wiki/%E8%88%AA%E6%B5%B7" title="航海">航海</a>和<a href="/wiki/%E5%9C%B0%E5%9B%BE%E5%AD%A6" title="地图学">测绘</a>等领域中。它们有重要的数学性质而在今天仍在广泛使用中。 </p> <div class="mw-heading mw-heading2"><h2 id="特殊底数"><span id=".E7.89.B9.E6.AE.8A.E5.BA.95.E6.95.B0"></span>特殊底数</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=8" title="编辑章节：特殊底数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>最常用做底数的是<a href="/wiki/E_(%E6%95%B0%E5%AD%A6%E5%B8%B8%E6%95%B0)" title="E (数学常数)"><i><big>e</big></i></a>、10和2。在<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90" title="数学分析">数学分析</a>中，以<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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>为底对数很常见。另一方面，以10为底对数在<a href="/wiki/%E5%8D%81%E8%BF%9B%E5%88%B6" title="十进制">十进制</a>表示法中，手工计算很容易：<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><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> <mn>10</mn> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mn>10</mn> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <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/280b4786e1799bef8cac9262a1bd71be9190db39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.234ex; height:2.676ex;" alt="{\displaystyle \log _{10}10x=\log _{10}10+\log _{10}x=1+\log _{10}x.\ }"></span></dd></dl> <p>所以<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}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> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f174fa0b73f4f350aff3b5d44cb831dcabfcea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.565ex; height:2.676ex;" alt="{\displaystyle \log _{10}x}"></span>表示正整数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>的位数：数字的十进制位数是严格大于<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}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> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f174fa0b73f4f350aff3b5d44cb831dcabfcea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.565ex; height:2.676ex;" alt="{\displaystyle \log _{10}x}"></span>的最小的整数。例如<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}1430\approx 3.15}"> <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> <mn>1430</mn> <mo>≈</mo> <mn>3.15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}1430\approx 3.15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85f47ca3e42eedf3050bba376437eaef1d6d918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.118ex; height:2.676ex;" alt="{\displaystyle \log _{10}1430\approx 3.15}"></span>，下一个整数是4，即1430的位数。 </p><p>以2为底的对数常用于计算机科学，因为计算机中二进制很普及。当然上面的算法也可推广到二进制：严格大于<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}x}"> <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> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3cea09debff9219b7ac089e9b0ff0c4f411fa60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.743ex; height:2.676ex;" alt="{\displaystyle \log _{2}x}"></span>的最小整数是<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>在二进制下的位数。事实上经由简单推导即可得知，floor(log<sub>p</sub>x)+1 得到<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>在<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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>进制下的位数：若<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>在<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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>进制下有<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>位，则<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 p^{n-1}\leq x<p^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≤</mo> <mi>x</mi> <mo><</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{n-1}\leq x<p^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e913db947ff2a86a991775bd1db14ab9756c7488" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.492ex; height:3.009ex;" alt="{\displaystyle p^{n-1}\leq x<p^{n}}"></span>；而<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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>是不小于 2 的正整数导致以其为底的<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 _{p}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{p}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d965f621a81d44ec10a882ef746052fb3b00a519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.748ex; height:3.009ex;" alt="{\displaystyle \log _{p}x}"></span>是增函数，故三边取对数得<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1\leq \log _{p}x<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>≤</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1\leq \log _{p}x<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/432341ecf0247eb7d8f8979cb1f72b45712f3884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.737ex; height:3.009ex;" alt="{\displaystyle n-1\leq \log _{p}x<n}"></span>，取下整正好得到<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span>。 </p><p>下表列出了这些底数的常用的对数符号以及他们所使用的领域。许多学科都写<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4157d3b51ac7b147fca145d431d58ec92abc1f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.111ex; height:2.843ex;" alt="{\displaystyle \log(x)}"></span>来代替<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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cad5f91f24392d04f1c9324d7154401e1ce3bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.048ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x)}"></span>，而<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>的值根据前后文可以确定。记号<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{b}\log(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{b}\log(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a97559ccab0584c2b9efe24f5efdc3d3e07891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.435ex; height:3.176ex;" alt="{\displaystyle ^{b}\log(x)}"></span>也出现过。<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>“ISO表示法”（<span class="ilh-all" data-orig-title="ISO 31-11" data-lang-code="en" data-lang-name="英语" data-foreign-title="ISO 31-11"><span class="ilh-page"><a href="/w/index.php?title=ISO_31-11&action=edit&redlink=1" class="new" title="ISO 31-11（页面不存在）">ISO 31-11</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/ISO_31-11" class="extiw" title="en:ISO 31-11"><span lang="en" dir="auto">ISO 31-11</span></a></span>）</span></span>）一列指定了<a href="/wiki/%E5%9C%8B%E9%9A%9B%E6%A8%99%E6%BA%96%E5%8C%96%E7%B5%84%E7%B9%94" title="國際標準化組織">ISO</a>推荐的表示方法。<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>6<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">底数<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> </th> <th scope="col"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777b35d8d91f445f98286c9619003039e28dfe94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.626ex; height:2.676ex;" alt="{\displaystyle \log _{b}x}"></span>的名称 </th> <th scope="col">ISO表示法 </th> <th scope="col">其它的表示方法 </th> <th scope="col">适用领域 </th></tr> <tr> <th scope="row">2 </th> <td><a href="/wiki/%E4%BA%8C%E8%BF%9B%E5%88%B6%E5%AF%B9%E6%95%B0" class="mw-redirect" title="二进制对数">二進制對數</a> </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 \operatorname {lb} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>lb</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {lb} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13da57b305fe2ac95b3106988087b4bea77491ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \operatorname {lb} x}"></span><sup id="cite_ref-gullberg_11-0" class="reference"><a href="#cite_note-gullberg-11"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d453de713a8c45f7bf99108752531ed7d6dd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.689ex; height:2.509ex;" alt="{\displaystyle \log x}"></span>、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {lg} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>lg</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {lg} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/951dd7c01b62be5e39fb5c81813ae4d8ce40ec9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.526ex; height:2.509ex;" alt="{\displaystyle \operatorname {lg} x}"></span> </td> <td>计算机科学、<a href="/wiki/%E4%BF%A1%E6%81%AF%E8%AE%BA" title="信息论">信息论</a>、数学 </td></tr> <tr> <th scope="row"><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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> </th> <td><a href="/wiki/%E8%87%AA%E7%84%B6%E5%AF%B9%E6%95%B0" class="mw-redirect" title="自然对数">自然对数</a> </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 \ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed172b0f5195382a3500c713941f945ad4db3898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \ln x}"></span><sup id="cite_ref-adaa_15-0" class="reference"><a href="#cite_note-adaa-15"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d453de713a8c45f7bf99108752531ed7d6dd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.689ex; height:2.509ex;" alt="{\displaystyle \log x}"></span><br />（用于数学和许多<a href="/wiki/%E7%A8%8B%E5%BA%8F%E8%AE%BE%E8%AE%A1%E8%AF%AD%E8%A8%80" class="mw-redirect" title="程序设计语言">程序设计语言</a><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup>） </td> <td>数学分析、物理学、化学<br /><a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%AD%A6" title="统计学">统计学</a>、<a href="/wiki/%E7%BB%8F%E6%B5%8E%E5%AD%A6" title="经济学">经济学</a>和其它工程领域 </td></tr> <tr> <th scope="row">10 </th> <td><a href="/wiki/%E5%B8%B8%E7%94%A8%E5%B0%8D%E6%95%B8" title="常用對數">常用对数</a> </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 \operatorname {lg} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>lg</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {lg} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/951dd7c01b62be5e39fb5c81813ae4d8ce40ec9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.526ex; height:2.509ex;" alt="{\displaystyle \operatorname {lg} x}"></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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d453de713a8c45f7bf99108752531ed7d6dd05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.689ex; height:2.509ex;" alt="{\displaystyle \log x}"></span><br />（用于工程学、生物学、天文学） </td> <td>多种<a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%AD%A6" title="工程学">工程学</a>领域（见<a href="/wiki/%E5%88%86%E8%B4%9D" class="mw-redirect" title="分贝">分贝</a>）、<br />对数<a href="/wiki/%E6%95%B0%E5%AD%A6%E7%94%A8%E8%A1%A8" title="数学用表">表</a>、手持式<a href="/wiki/%E8%AE%A1%E7%AE%97%E5%99%A8" class="mw-redirect" title="计算器">计算器</a>、 <a href="/wiki/%E5%85%89%E8%B0%B1%E5%AD%A6" title="光谱学">光谱学</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="底数变换"><span id=".E5.BA.95.E6.95.B0.E5.8F.98.E6.8D.A2"></span>底数变换</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=9" title="编辑章节：底数变换"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>尽管有很多有用的恒等式，对计算器最重要的是找到不是建造于计算器内的底数（通常是<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 _{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26f8281388ea19ac65fb6d3764c95e8d2546e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.97ex; height:2.676ex;" alt="{\displaystyle \log _{e}}"></span>和<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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1be2659542850a5a4def5da1ca2bb402fbb4e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.848ex; height:2.676ex;" alt="{\displaystyle \log _{10}}"></span>）的其他底数的对数。要使用其他底数<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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>找到底数<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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>的对数: </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 _{\alpha }x={\frac {\log _{\beta }x}{\log _{\beta }\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }x={\frac {\log _{\beta }x}{\log _{\beta }\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bced6805fafd1d49c52457b5fecdbe69cad1a388" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.928ex; height:6.843ex;" alt="{\displaystyle \log _{\alpha }x={\frac {\log _{\beta }x}{\log _{\beta }\alpha }}}"></span>。</dd></dl> <p>此外，这个结果蕴涵了所有对数函数（任意底数）都是相互类似的。所以用计算器计算对134217728底数2的对数: </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}134217728={\frac {\ln 134217728}{\ln 2}}={\frac {27\ln 2}{\ln 2}}=27}"> <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> <mn>134217728</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>134217728</mn> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>27</mn> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>27</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}134217728={\frac {\ln 134217728}{\ln 2}}={\frac {27\ln 2}{\ln 2}}=27}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cb50db4ac11fb5bb3bbb8606638634cc28ef680" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.157ex; height:5.343ex;" alt="{\displaystyle \log _{2}134217728={\frac {\ln 134217728}{\ln 2}}={\frac {27\ln 2}{\ln 2}}=27}"></span>。</dd></dl> <div class="mw-heading mw-heading2"><h2 id="对数的用途"><span id=".E5.AF.B9.E6.95.B0.E7.9A.84.E7.94.A8.E9.80.94"></span>对数的用途</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=10" title="编辑章节：对数的用途"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>对数对解幂是未知的方程是有用的。它们有简单的<a href="/wiki/%E5%AF%BC%E6%95%B0" title="导数">导数</a>，所以它们经常用在解<a href="/wiki/%E7%A7%AF%E5%88%86" title="积分">积分</a>中。对数是三个相关的函数中的一个。在等式<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}=x}"> <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> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{n}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d38c47f05d9c8c70fe924862c2b16f0e909599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.644ex; height:2.343ex;" alt="{\displaystyle b^{n}=x}"></span>中，<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>可以从<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>次<a href="/wiki/%E6%96%B9%E6%A0%B9" title="方根">方根</a>，<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i>从<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>的<i><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span></i>底数的对数，<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span></i>从<i><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span></i>的<i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></i>次的<a href="/wiki/%E5%B9%82" class="mw-redirect" title="幂">幂</a>来确定。参见<a href="/wiki/%E5%AF%B9%E6%95%B0%E6%81%92%E7%AD%89%E5%BC%8F" title="对数恒等式">对数恒等式</a>得到掌控对数函数的一些规则。 </p> <div class="mw-heading mw-heading3"><h3 id="简便计算"><span id=".E7.AE.80.E4.BE.BF.E8.AE.A1.E7.AE.97"></span>简便计算</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=11" title="编辑章节：简便计算"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>对数把注意力从平常的数转移到了幂。只要使用相同的底数，就会使特定运算更容易： </p> <table class="wikitable"> <tbody><tr> <th>数的运算</th> <th>幂的运算</th> <th>对数恒等式 </th></tr> <tr> <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 \,xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced755ce5a7adccefc12d2702d1eb42ccf1f126c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.872ex; height:2.009ex;" alt="{\displaystyle \,xy}"></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 \,m+n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>m</mi> <mo>+</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,m+n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d33c44c644c39645b9b20c9d2deeeba5cb6629c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.663ex; height:2.176ex;" alt="{\displaystyle \,m+n}"></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 _{\theta }xy=\log _{\theta }x+\log _{\theta }y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mi>y</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\log _{\theta }xy=\log _{\theta }x+\log _{\theta }y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7409a2e04e2b4f3bda2af2f4a3e6aed664c89f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.383ex; height:2.676ex;" alt="{\displaystyle \,\log _{\theta }xy=\log _{\theta }x+\log _{\theta }y}"></span> </td></tr> <tr> <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 {\frac {x}{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/187d36a20476c4ccaaae85662613087bc4c67d35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.166ex; height:5.176ex;" alt="{\displaystyle {\frac {x}{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 \,m-n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>m</mi> <mo>−</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,m-n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4441e42d12a3c399c5ee4301f90b0508512838f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.663ex; height:2.176ex;" alt="{\displaystyle \,m-n}"></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 _{\theta }{\frac {x}{y}}=\log _{\theta }x-\log _{\theta }y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\theta }{\frac {x}{y}}=\log _{\theta }x-\log _{\theta }y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f20ccd3a70a11fc6d54be5c6571f9528d0ab3247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.676ex; height:5.176ex;" alt="{\displaystyle \log _{\theta }{\frac {x}{y}}=\log _{\theta }x-\log _{\theta }y}"></span> </td></tr> <tr> <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 \,x^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,x^{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8844106923a115ce311618a402072d9ea386da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.766ex; height:2.343ex;" alt="{\displaystyle \,x^{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 \,mn}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>m</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,mn}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ec2bd238cb0cd1ab9ddaa9dfdcecf4704c8b05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.822ex; height:1.676ex;" alt="{\displaystyle \,mn}"></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 _{\theta }x^{y}=y\log _{\theta }x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mo>=</mo> <mi>y</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\log _{\theta }x^{y}=y\log _{\theta }x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84c35fe6e663a248a1ab7dd7a5dc54e8da0796c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.461ex; height:2.843ex;" alt="{\displaystyle \,\log _{\theta }x^{y}=y\log _{\theta }x}"></span> </td></tr> <tr> <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 {\sqrt[{y}]{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{y}]{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071bdbfd539611cb6a2e4a5755b6838b63196ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.266ex; height:3.009ex;" alt="{\displaystyle {\sqrt[{y}]{x}}}"></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 {\frac {m}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d48d87468620ad6c70385ddd0d024577ccb559e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.877ex; height:4.676ex;" alt="{\displaystyle {\frac {m}{n}}}"></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 _{\theta }{\sqrt[{y}]{x}}={\frac {\log _{\theta }x}{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </mroot> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mrow> <mi>y</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\theta }{\sqrt[{y}]{x}}={\frac {\log _{\theta }x}{y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9a87f00ad760db11bcd291e05f07877f0247c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.254ex; height:6.009ex;" alt="{\displaystyle \log _{\theta }{\sqrt[{y}]{x}}={\frac {\log _{\theta }x}{y}}}"></span> </td></tr></tbody></table> <p>这些关系使在两个数上的这种运算更快，<style data-mw-deduplicate="TemplateStyles:r83946278">.mw-parser-output .template-facttext{background-color:var(--background-color-neutral,#eaecf0);color:inherit;margin:-.3em 0;padding:.3em 0}</style><mark class="template-facttext" title="需要提供文献来源">在加法<a href="/wiki/%E8%AE%A1%E7%AE%97%E5%99%A8" class="mw-redirect" title="计算器">计算器</a>出现之前正确的使用对数是基本技能。</mark><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求。">[來源請求]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="群论"><span id=".E7.BE.A4.E8.AE.BA"></span>群论</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=12" title="编辑章节：群论"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>从纯数学的观点来看，恒等式： <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 _{\alpha }\mathrm {M} \mathrm {N} =\log _{\alpha }\mathrm {M} +\log _{\alpha }\mathrm {N} \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mo>+</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\alpha }\mathrm {M} \mathrm {N} =\log _{\alpha }\mathrm {M} +\log _{\alpha }\mathrm {N} \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ed84ac87c1910dd4e711cc14895dc5d00d22b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.327ex; width:27.556ex; height:2.676ex;" alt="{\displaystyle \log _{\alpha }\mathrm {M} \mathrm {N} =\log _{\alpha }\mathrm {M} +\log _{\alpha }\mathrm {N} \!}"></span>，在两种意义上是基本的。首先，其他3个算术性质可以从它得出。进一步的，它表达了在正实数的<b><span style="font-color:'#Ff0000'">乘法群</span></b>和所有实数的<b><span style="font-color:'#0000ff'">加法群</span></b>之间的<a href="/wiki/%E5%90%8C%E6%9E%84" title="同构">同构</a>。 </p><p>对数函数是从正实数的乘法群到实数的加法群的唯一连续同构。 </p> <div class="mw-heading mw-heading2"><h2 id="复对数"><span id=".E5.A4.8D.E5.AF.B9.E6.95.B0"></span>复对数</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=13" title="编辑章节：复对数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E8%A4%87%E5%B0%8D%E6%95%B8" title="複對數">複對數</a></div> <p>复对数计算公式： </p> <div style="overflow-x: auto"> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{c+di}(a+bi)={\frac {\ln \left(a^{2}+b^{2}\right)\cdot \ln \left(c^{2}+d^{2}\right)+4\left(\arctan {\frac {b}{a}}+2k\pi \right)\left(\arctan {\frac {d}{c}}+2n\pi \right)+\left[2\left(\arctan {\frac {b}{a}}+2k\pi \right)\ln \left(c^{2}+d^{2}\right)-2\left(\arctan {\frac {d}{c}}+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]i}{\ln ^{2}\left(c^{2}+d^{2}\right)+4\left(\arctan {\frac {d}{c}}+2n\pi \right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>c</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>c</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mi>i</mi> </mrow> <mrow> <msup> <mi>ln</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>4</mn> <msup> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>c</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{c+di}(a+bi)={\frac {\ln \left(a^{2}+b^{2}\right)\cdot \ln \left(c^{2}+d^{2}\right)+4\left(\arctan {\frac {b}{a}}+2k\pi \right)\left(\arctan {\frac {d}{c}}+2n\pi \right)+\left[2\left(\arctan {\frac {b}{a}}+2k\pi \right)\ln \left(c^{2}+d^{2}\right)-2\left(\arctan {\frac {d}{c}}+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]i}{\ln ^{2}\left(c^{2}+d^{2}\right)+4\left(\arctan {\frac {d}{c}}+2n\pi \right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa379461024fefda2294fc1c0d0e26aa5411e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:154.477ex; height:10.676ex;" alt="{\displaystyle \log _{c+di}(a+bi)={\frac {\ln \left(a^{2}+b^{2}\right)\cdot \ln \left(c^{2}+d^{2}\right)+4\left(\arctan {\frac {b}{a}}+2k\pi \right)\left(\arctan {\frac {d}{c}}+2n\pi \right)+\left[2\left(\arctan {\frac {b}{a}}+2k\pi \right)\ln \left(c^{2}+d^{2}\right)-2\left(\arctan {\frac {d}{c}}+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]i}{\ln ^{2}\left(c^{2}+d^{2}\right)+4\left(\arctan {\frac {d}{c}}+2n\pi \right)^{2}}}}"></span></dd></dl> </div> <div style="overflow-x: auto"> <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 (a+bi)^{\left(c+di\right)}=e^{{\frac {c}{2}}\ln \left(a^{2}+b^{2}\right)-\left(d+2n\pi \right)\left(\arctan {\frac {b}{a}}+2k\pi \right)}\left\{\cos \left[c\left(\arctan {\frac {b}{a}}+2k\pi \right)+{\frac {1}{2}}\left(d+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]+i\sin \left[c\left(\arctan {\frac {b}{a}}+2k\pi \right)+{\frac {1}{2}}\left(d+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mrow> <mo>{</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mi>π</mi> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+bi)^{\left(c+di\right)}=e^{{\frac {c}{2}}\ln \left(a^{2}+b^{2}\right)-\left(d+2n\pi \right)\left(\arctan {\frac {b}{a}}+2k\pi \right)}\left\{\cos \left[c\left(\arctan {\frac {b}{a}}+2k\pi \right)+{\frac {1}{2}}\left(d+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]+i\sin \left[c\left(\arctan {\frac {b}{a}}+2k\pi \right)+{\frac {1}{2}}\left(d+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4afb62a51ada41cb1c2cedbd6aaf10b03eb9fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:158.684ex; height:6.176ex;" alt="{\displaystyle (a+bi)^{\left(c+di\right)}=e^{{\frac {c}{2}}\ln \left(a^{2}+b^{2}\right)-\left(d+2n\pi \right)\left(\arctan {\frac {b}{a}}+2k\pi \right)}\left\{\cos \left[c\left(\arctan {\frac {b}{a}}+2k\pi \right)+{\frac {1}{2}}\left(d+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]+i\sin \left[c\left(\arctan {\frac {b}{a}}+2k\pi \right)+{\frac {1}{2}}\left(d+2n\pi \right)\ln \left(a^{2}+b^{2}\right)\right]\right\}}"></span></dd></dl> </div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}\arctan 0={\pi },&{\mbox{for }}a<0\!\,\\\arctan 0=0,&{\mbox{for }}a>0\!\,\\\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for </mtext> </mstyle> </mrow> <mi>a</mi> <mo><</mo> <mn>0</mn> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mtd> </mtr> <mtr> <mtd> <mi>arctan</mi> <mo>⁡</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for </mtext> </mstyle> </mrow> <mi>a</mi> <mo>></mo> <mn>0</mn> <mspace width="negativethinmathspace" /> <mspace width="thinmathspace" /> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\arctan 0={\pi },&{\mbox{for }}a<0\!\,\\\arctan 0=0,&{\mbox{for }}a>0\!\,\\\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeeab53b05de5d5c6b519d9ef5d94c33bab47c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.767ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\arctan 0={\pi },&{\mbox{for }}a<0\!\,\\\arctan 0=0,&{\mbox{for }}a>0\!\,\\\end{cases}}}"></span></dd></dl> <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 \mathbb {Z} =\{k,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} =\{k,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f192ef4e19931e1aa0ab001f59829e5471d39d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.614ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} =\{k,n\}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="微积分"><span id=".E5.BE.AE.E7.A7.AF.E5.88.86"></span>微积分</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=14" title="编辑章节：微积分"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%AF%B9%E6%95%B0%E5%BE%AE%E5%88%86%E6%B3%95" title="对数微分法">对数微分法</a></div> <p>自然对数函数的<a href="/wiki/%E5%AF%BC%E6%95%B0" title="导数">导数</a>是 </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 {\rm {d}}{{\rm {d}}x}}\ln \left|x\right|={\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}\ln \left|x\right|={\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ecceea8e9bd9b87b105b8f4fd2e90c9dabf545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.672ex; height:5.509ex;" alt="{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}\ln \left|x\right|={\frac {1}{x}}}"></span>。</dd></dl> <p>通过应用换底规则，其他底数的导数是 </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 {\rm {d}}{{\rm {d}}x}}\log _{b}x={\frac {\rm {d}}{{\rm {d}}x}}{\frac {\ln x}{\ln b}}={\frac {1}{x\ln b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}\log _{b}x={\frac {\rm {d}}{{\rm {d}}x}}{\frac {\ln x}{\ln b}}={\frac {1}{x\ln b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4597bb84f98d13127d9e8f44f4c01d69b2e8bd28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.496ex; height:5.509ex;" alt="{\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}\log _{b}x={\frac {\rm {d}}{{\rm {d}}x}}{\frac {\ln x}{\ln b}}={\frac {1}{x\ln b}}}"></span>。</dd></dl> <p>自然对数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7df974798a70099a373bf0039df34200c9a622e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.043ex; height:2.176ex;" alt="{\displaystyle \ln x\,}"></span>的<a href="/wiki/%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86" title="不定积分">不定积分</a>是 </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 \int \ln x\,{\rm {d}}x=x\ln x-x+C,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>C</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \ln x\,{\rm {d}}x=x\ln x-x+C,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeaa74bda355c33d6f45b3f44d351dd04538ad87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.141ex; height:5.676ex;" alt="{\displaystyle \int \ln x\,{\rm {d}}x=x\ln x-x+C,}"></span></dd></dl> <p>而其他底数对数的<a href="/wiki/%E4%B8%8D%E5%AE%9A%E7%A7%AF%E5%88%86" title="不定积分">不定积分</a>是 </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 \int \log _{b}x\,{\rm {d}}x=x\log _{b}x-{\frac {x}{\ln b}}+C=x\log _{b}{\frac {x}{e}}+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mi>x</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>e</mi> </mfrac> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \log _{b}x\,{\rm {d}}x=x\log _{b}x-{\frac {x}{\ln b}}+C=x\log _{b}{\frac {x}{e}}+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1271993c0818d1cdf6dd2e5c191cf568aea6f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.149ex; height:5.676ex;" alt="{\displaystyle \int \log _{b}x\,{\rm {d}}x=x\log _{b}x-{\frac {x}{\ln b}}+C=x\log _{b}{\frac {x}{e}}+C}"></span>。</dd></dl> <div class="mw-heading mw-heading2"><h2 id="计算自然对数的级数"><span id=".E8.AE.A1.E7.AE.97.E8.87.AA.E7.84.B6.E5.AF.B9.E6.95.B0.E7.9A.84.E7.BA.A7.E6.95.B0"></span>计算自然对数的级数</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=15" title="编辑章节：计算自然对数的级数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>有一些<a href="/wiki/%E7%BA%A7%E6%95%B0" title="级数">级数</a>用来计算自然对数。<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>最简单和低效的是: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln z=\sum _{n=1}^{\infty }{\frac {-{(-1)}^{n}}{n}}(z-1)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mi>n</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln z=\sum _{n=1}^{\infty }{\frac {-{(-1)}^{n}}{n}}(z-1)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0662729893116e5d24ce74938380f71d97118fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.017ex; height:6.843ex;" alt="{\displaystyle \ln z=\sum _{n=1}^{\infty }{\frac {-{(-1)}^{n}}{n}}(z-1)^{n}}"></span>当<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 |z-1|<1\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z-1|<1\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc54065ebae979a740f0b66a45979029e59adf36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.218ex; width:10.476ex; height:2.843ex;" alt="{\displaystyle |z-1|<1\!}"></span>。</dd></dl> <p>下做推导： </p><p>由 </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 {1}{1-x}}=1+x+x^{2}+x^{3}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f1a5bf8703c75966ad31a998b6e8a03f6d7cb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.612ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\cdots }"></span>。</dd></dl> <p>在两边积分得到 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\ln(1-x)=x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ln(1-x)=x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a035e2933963431800629fd56e1d51bbfb986d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.389ex; height:5.676ex;" alt="{\displaystyle -\ln(1-x)=x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+\cdots }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(1-x)=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}-\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(1-x)=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}-\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc819d94e6b72a26895b6b934b96586fabe33d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.062ex; height:5.676ex;" alt="{\displaystyle \ln(1-x)=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}-\cdots }"></span>。</dd></dl> <p>设<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 z=1-x\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=1-x\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f4b595c16d056854b4cb568d5ce40d94fe0e28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; margin-right: -0.271ex; width:9.403ex; height:2.343ex;" alt="{\displaystyle z=1-x\!}"></span>并因此<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=-(z-1)\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=-(z-1)\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4cadfa4b14fcd96163337256ddab33d346d4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.166ex; width:12.916ex; height:2.843ex;" alt="{\displaystyle x=-(z-1)\!}"></span>，得到 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln z=(z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln z=(z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88176f3d470e312ef7df3b8bed40ac70dade950c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.87ex; height:5.843ex;" alt="{\displaystyle \ln z=(z-1)-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots }"></span></dd></dl> <p>更有效率的级数是基於<a href="/wiki/%E5%8F%8D%E5%8F%8C%E6%9B%B2%E5%87%BD%E6%95%B0" title="反双曲函数">反雙曲函數</a>的 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln z=2\sum _{n=0}^{\infty }{\frac {1}{2n+1}}\left({\frac {z-1}{z+1}}\right)^{2n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln z=2\sum _{n=0}^{\infty }{\frac {1}{2n+1}}\left({\frac {z-1}{z+1}}\right)^{2n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed51c8cac9f74b4bfa5b27b895d3b400b32cdfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.69ex; height:7.009ex;" alt="{\displaystyle \ln z=2\sum _{n=0}^{\infty }{\frac {1}{2n+1}}\left({\frac {z-1}{z+1}}\right)^{2n+1}}"></span></dd></dl> <p>对带有正实部的<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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>。 </p><p>推导：代换<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae55e66aeffc525917eed885b4b753ba5a7f8b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}"></span>为<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>，得到 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/462c979aa1d56163e98c6226c383af97b8d16a45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.254ex; height:5.676ex;" alt="{\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots }"></span>。</dd></dl> <p>做减法，得到 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\frac {1+x}{1-x}}=\ln(1+x)-\ln(1-x)=2x+2{\frac {x^{3}}{3}}+2{\frac {x^{5}}{5}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln {\frac {1+x}{1-x}}=\ln(1+x)-\ln(1-x)=2x+2{\frac {x^{3}}{3}}+2{\frac {x^{5}}{5}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890ef721bba4ffd4df8569f2d5951e1daae5b9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:58.196ex; height:5.843ex;" alt="{\displaystyle \ln {\frac {1+x}{1-x}}=\ln(1+x)-\ln(1-x)=2x+2{\frac {x^{3}}{3}}+2{\frac {x^{5}}{5}}+\cdots }"></span>。</dd></dl> <p>设<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 z={\frac {1+x}{1-x}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\frac {1+x}{1-x}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd5a843af2c83d15cdddc6e65bdc8b78e2e667c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.108ex; width:10.077ex; height:5.343ex;" alt="{\displaystyle z={\frac {1+x}{1-x}}\!}"></span>并因此<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={\frac {z-1}{z+1}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {z-1}{z+1}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/589c011c1ce76635a45ae599ba03e09f16c11f55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.108ex; width:10.077ex; height:5.343ex;" alt="{\displaystyle x={\frac {z-1}{z+1}}\!}"></span>，得到 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln z=2\left[{\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln z=2\left[{\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52341a46f5631d30a9166d55d62bed6819a1d564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.747ex; height:7.509ex;" alt="{\displaystyle \ln z=2\left[{\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right]}"></span>。</dd></dl> <p>例如，应用这个级数于 </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 z={\frac {11}{9}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>9</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z={\frac {11}{9}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f27896e473d5091b99d4d385a3c14ec8b1f900b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.994ex; height:5.176ex;" alt="{\displaystyle z={\frac {11}{9}},}"></span></dd></dl> <p>得到 </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 {z-1}{z+1}}={\frac {{\frac {11}{9}}-1}{{\frac {11}{9}}+1}}={\frac {1}{10}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>9</mn> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>9</mn> </mfrac> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {z-1}{z+1}}={\frac {{\frac {11}{9}}-1}{{\frac {11}{9}}+1}}={\frac {1}{10}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05aae03e4301133b1f36eca5bab0fee14cb88521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:23.251ex; height:8.176ex;" alt="{\displaystyle {\frac {z-1}{z+1}}={\frac {{\frac {11}{9}}-1}{{\frac {11}{9}}+1}}={\frac {1}{10}},}"></span></dd></dl> <p>并因此 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln 1.{\dot {2}}={\frac {1}{5}}\left(1+{\frac {1}{3\cdot 100}}+{\frac {1}{5\cdot 10000}}+{\frac {1}{7\cdot 1000000}}+\cdots \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mn>1.</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>100</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>5</mn> <mo>⋅</mo> <mn>10000</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>7</mn> <mo>⋅</mo> <mn>1000000</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln 1.{\dot {2}}={\frac {1}{5}}\left(1+{\frac {1}{3\cdot 100}}+{\frac {1}{5\cdot 10000}}+{\frac {1}{7\cdot 1000000}}+\cdots \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2ff04e7e514d954ae5d69fe372d98d953efeb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.921ex; height:6.176ex;" alt="{\displaystyle \ln 1.{\dot {2}}={\frac {1}{5}}\left(1+{\frac {1}{3\cdot 100}}+{\frac {1}{5\cdot 10000}}+{\frac {1}{7\cdot 1000000}}+\cdots \right)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =0.2\cdot (1.0000000\dots +0.00{\dot {3}}+0.00002+0.000000{\dot {1}}4285{\dot {7}}+\cdots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>0.2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1.0000000</mn> <mo>⋯</mo> <mo>+</mo> <mn>0.00</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mn>0.00002</mn> <mo>+</mo> <mn>0.000000</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo>˙</mo> </mover> </mrow> </mrow> <mn>4285</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>7</mn> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mo>⋯</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =0.2\cdot (1.0000000\dots +0.00{\dot {3}}+0.00002+0.000000{\dot {1}}4285{\dot {7}}+\cdots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62e7cb6bc59ee0585b6a52b2389f7450b2e23f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.732ex; height:3.176ex;" alt="{\displaystyle =0.2\cdot (1.0000000\dots +0.00{\dot {3}}+0.00002+0.000000{\dot {1}}4285{\dot {7}}+\cdots )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =0.2\cdot 1.00335\cdots =0.200670\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>0.2</mn> <mo>⋅</mo> <mn>1.00335</mn> <mo>⋯</mo> <mo>=</mo> <mn>0.200670</mn> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =0.2\cdot 1.00335\cdots =0.200670\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/820a09cda7f59b1e7837ca6773086ce740fe1905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:32.829ex; height:2.176ex;" alt="{\displaystyle =0.2\cdot 1.00335\cdots =0.200670\cdots }"></span></dd></dl> <p>在这里我们在第一行的总和中提出了因数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7319d1751af9cc0419f3fba7803a4474f2bddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.161ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{10}}}"></span>。 </p><p>对于任何其他底数<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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>，我们使用 </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 _{\beta }x={\frac {\ln x}{\ln \beta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mrow> <mrow> <mi>ln</mi> <mo>⁡</mo> <mi>β</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{\beta }x={\frac {\ln x}{\ln \beta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f6b9d59ed7f97f23d2778c9ed45c89e2df4a7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.456ex; height:5.843ex;" alt="{\displaystyle \log _{\beta }x={\frac {\ln x}{\ln \beta }}}"></span>。</dd></dl> <div id="noteTA-f3e2a619" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="IT"></div></div></div> <div class="mw-heading mw-heading2"><h2 id="计算机"><span id=".E8.AE.A1.E7.AE.97.E6.9C.BA"></span>计算机</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=16" title="编辑章节：计算机"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>多数<a href="/wiki/%E8%AE%A1%E7%AE%97%E6%9C%BA%E8%AF%AD%E8%A8%80" title="计算机语言">计算机语言</a>把<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4157d3b51ac7b147fca145d431d58ec92abc1f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.111ex; height:2.843ex;" alt="{\displaystyle \log(x)}"></span>用做自然对数，而常用对数典型的指示为log10(x)。参数和返回值典型的是浮点数据类型。 </p><p>因为参数是浮点数，可以有用的做如下考虑： </p><p>浮点数值<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>被表示为尾数<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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>和指数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>所形成的 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=m2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>m</mi> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=m2^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9907470c4afd3e6320dbfdc41648eda3a166ee82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.849ex; height:2.343ex;" alt="{\displaystyle x=m2^{n}}"></span>。</dd></dl> <p>因此 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)=\ln(m)+n\ln(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>n</mi> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)=\ln(m)+n\ln(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789e1d97647a7f6b7570f9b84ecbb11899227dde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.499ex; height:2.843ex;" alt="{\displaystyle \ln(x)=\ln(m)+n\ln(2)}"></span>。</dd></dl> <p>所以，替代计算<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df055b8e294310e6785701c1c67105e109191d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \ln(x)}"></span>，我们计算对某个<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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(m)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edb0a1647131eefc54279a367be1e16c495b095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle \ln(m)}"></span>使得<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 1\leq m\leq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq m\leq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a56cb3badaa89f2c3fa80117ccc30d782eab4e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.562ex; height:2.343ex;" alt="{\displaystyle 1\leq m\leq 2}"></span>。有在这个范围内的<i><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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span></i>意味着值<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {m-1}{m+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {m-1}{m+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b7187d95523df84284f89a3859c638adabfe03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.308ex; height:5.343ex;" alt="{\displaystyle u={\frac {m-1}{m+1}}}"></span>总是在范围<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq u<{\frac {1}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>u</mi> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq u<{\frac {1}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57da81d42357516a64ca0cd8874e6780061201b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.688ex; height:5.176ex;" alt="{\displaystyle 0\leq u<{\frac {1}{3}}}"></span>内。某些机器使用在范围<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.5\leq m<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.5</mn> <mo>≤</mo> <mi>m</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.5\leq m<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7966437d0a88bc4b47cd0e6b513f9c41ffaa01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.372ex; height:2.343ex;" alt="{\displaystyle 0.5\leq m<1}"></span>内的尾数，并且在这个情况下<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span>的值将在范围<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {1}{3}}<u\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo><</mo> <mi>u</mi> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{3}}<u\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b207964629bdb6723e9a0b06e2eac1d74ec4a5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.496ex; height:5.176ex;" alt="{\displaystyle -{\frac {1}{3}}<u\leq 0}"></span>内。在任何一种情况下，这个级数都是更容易计算的。 </p> <div class="mw-heading mw-heading2"><h2 id="一般化"><span id=".E4.B8.80.E8.88.AC.E5.8C.96"></span>一般化</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=17" title="编辑章节：一般化"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>普通的正实数的对数一般化为负数和<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>参数，尽管它是<a href="/wiki/%E5%A4%9A%E5%80%BC%E5%87%BD%E6%95%B0" title="多值函数">多值函数</a>，需要终止在<a href="/w/index.php?title=%E5%88%86%E6%94%AF%E7%82%B9&action=edit&redlink=1" class="new" title="分支点（页面不存在）">分支点</a>0上的分支切割，来制作一个普通函数或主分支。复数<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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>的（底数<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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>）的对数是复数<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(\left\vert z\right\vert )+i\arg(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow> <mo>|</mo> <mi>z</mi> <mo>|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>i</mi> <mi>arg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(\left\vert z\right\vert )+i\arg(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891bcd5e3bdd3123f68ab0d6fefab0ef3c4e0715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.294ex; height:2.843ex;" alt="{\displaystyle \ln(\left\vert z\right\vert )+i\arg(z)}"></span>，这裡的<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert z\right\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi>z</mi> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\vert z\right\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09d238ddf8dd3a23b8355dccaf1b6fe1b39a68a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.382ex; height:2.843ex;" alt="{\displaystyle \left\vert z\right\vert }"></span>是<i><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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span></i>的<a href="/wiki/%E8%A4%87%E6%95%B8_(%E6%95%B8%E5%AD%B8)#複數平面" class="mw-redirect" title="複數 (數學)">模</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 \arg(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arg(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8bb9d3cf9c1a82f1666e4e72b6702b0cee4b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.134ex; height:2.843ex;" alt="{\displaystyle \arg(z)}"></span>是<a href="/wiki/%E8%A4%87%E6%95%B8_(%E6%95%B8%E5%AD%B8)#複數平面" class="mw-redirect" title="複數 (數學)">辐角</a>，而<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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>是<a href="/wiki/%E8%99%9A%E5%8D%95%E4%BD%8D" class="mw-redirect" title="虚单位">虚单位</a>；详情参见<a href="/wiki/%E8%A4%87%E5%B0%8D%E6%95%B8" title="複對數">复对数</a>。 </p><p><a href="/wiki/%E7%A6%BB%E6%95%A3%E5%AF%B9%E6%95%B0" title="离散对数">离散对数</a>是在<a href="/wiki/%E6%9C%89%E9%99%90%E7%BE%A4" title="有限群">有限群</a>理论中的相关概念。它涉及到解方程<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}=x}"> <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> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{n}=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d38c47f05d9c8c70fe924862c2b16f0e909599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.644ex; height:2.343ex;" alt="{\displaystyle b^{n}=x}"></span>，这裡的<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>和<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>是这个群的元素，而<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>是指定在群运算上的幂。对于某些有限群，据信离散对数是非常难计算的，而离散指数非常容易。这种不对称性可用于<a href="/wiki/%E5%85%AC%E5%BC%80%E5%AF%86%E9%92%A5%E5%8A%A0%E5%AF%86" title="公开密钥加密">公开密钥加密</a>。 </p><p><a href="/wiki/%E7%9F%A9%E9%98%B5%E5%AF%B9%E6%95%B0" title="矩阵对数">矩阵对数</a>是<a href="/wiki/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" title="矩阵指数">矩阵指数</a>的反函数。 </p><p>对于不等于1的每个正数<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>，函数<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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{b}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cad5f91f24392d04f1c9324d7154401e1ce3bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.048ex; height:2.843ex;" alt="{\displaystyle \log _{b}(x)}"></span>是从在乘法下的正实数的<a href="/wiki/%E7%BE%A4" title="群">群</a>到在加法下（所有）实数的群的<a href="/wiki/%E5%90%8C%E6%9E%84" title="同构">同构</a>。它们是唯一的连续的这种同构。对数函数可以扩展为在乘法下正实数的<a href="/wiki/%E6%8B%93%E6%89%91%E7%A9%BA%E9%97%B4" title="拓扑空间">拓扑空间</a>的<a href="/wiki/%E5%93%88%E5%B0%94%E6%B5%8B%E5%BA%A6" title="哈尔测度">哈尔测度</a>。 </p> <div class="mw-heading mw-heading2"><h2 id="对数表"><span id=".E5.AF.B9.E6.95.B0.E8.A1.A8"></span>对数表</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=18" title="编辑章节：对数表"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Abramowitz%26Stegun.page97.agr.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Abramowitz%26Stegun.page97.agr.jpg/400px-Abramowitz%26Stegun.page97.agr.jpg" decoding="async" width="400" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Abramowitz%26Stegun.page97.agr.jpg/600px-Abramowitz%26Stegun.page97.agr.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Abramowitz%26Stegun.page97.agr.jpg/800px-Abramowitz%26Stegun.page97.agr.jpg 2x" data-file-width="1200" data-file-height="791" /></a><figcaption>20世纪的<a href="/wiki/%E5%B8%B8%E7%94%A8%E5%B0%8D%E6%95%B8" title="常用對數">常用对数</a>表的一个实例。</figcaption></figure> <div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%AF%B9%E6%95%B0%E8%A1%A8" title="对数表">对数表</a></div> <p>在發明<a href="/wiki/%E8%AE%A1%E7%AE%97%E5%99%A8" class="mw-redirect" title="计算器">计算器</a>之前，使用对数意味着查<a href="/wiki/%E5%AF%B9%E6%95%B0%E8%A1%A8" title="对数表">对数表</a>，它必须手工建立。 </p> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=19" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%AF%B9%E6%95%B0%E6%81%92%E7%AD%89%E5%BC%8F" title="对数恒等式">对数恒等式</a></li> <li><a href="/wiki/%E8%87%AA%E7%84%B6%E5%AF%B9%E6%95%B0" class="mw-redirect" title="自然对数">自然对数</a></li> <li><a href="/wiki/%E5%B8%B8%E7%94%A8%E5%B0%8D%E6%95%B8" title="常用對數">常用对数</a></li> <li><a href="/wiki/%E7%A6%BB%E6%95%A3%E5%AF%B9%E6%95%B0" title="离散对数">离散对数</a></li> <li><a href="/wiki/%E8%8A%AE%E6%B0%8F%E5%9C%B0%E9%9C%87%E8%A7%84%E6%A8%A1" class="mw-redirect" title="芮氏地震规模">芮氏地震规模</a></li> <li><a href="/wiki/%E5%88%86%E8%B4%9D" class="mw-redirect" title="分贝">分贝</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="注释"><span id=".E6.B3.A8.E9.87.8A"></span>注释</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=20" title="编辑章节：注释"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="notelist" style="list-style-type: lower-alpha;"> <ol class="references"> <li id="cite_note-adaa-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-adaa_15-0">^</a></b></span> <span class="reference-text">一些数学家反对这种表示法。在他的1985年的自传中，<a href="/wiki/%E4%BF%9D%E7%BE%85%C2%B7%E5%93%88%E7%88%BE%E8%8E%AB%E6%96%AF" title="保羅·哈爾莫斯">保羅·哈爾莫斯</a>批评了这种表示法，称之为“幼稚的表示法”，他说没有一位数学家这么用过<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>。这种表示法是数学家<span class="ilh-all" data-orig-title="Irving Stringham" data-lang-code="en" data-lang-name="英语" data-foreign-title="Irving Stringham"><span class="ilh-page"><a href="/w/index.php?title=Irving_Stringham&action=edit&redlink=1" class="new" title="Irving Stringham（页面不存在）">Irving Stringham</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Irving_Stringham" class="extiw" title="en:Irving Stringham"><span lang="en" dir="auto">Irving Stringham</span></a></span>）</span></span>发明的<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">例如 <a href="/wiki/C%E8%AF%AD%E8%A8%80" title="C语言">C语言</a>、<a href="/wiki/Java" title="Java">Java语言</a>、<a href="/wiki/Haskell" title="Haskell">Haskell语言</a>和<a href="/wiki/BASIC" title="BASIC">BASIC语言</a>。</span> </li> </ol> </div> <div id="references-NoteFoot"><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">所有底数的对数函数都通过点（1,0），因为任何数的0次幂都是1（0除外），而底数 <i>β</i> 的函数通过点（<i>β</i> , 1），因为任何数的1次幂都是自身1。曲线接近 <i>y</i> 轴但永不触及它，因为 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>的<a href="/w/index.php?title=%E5%A5%87%E5%BC%82%E6%80%A7&action=edit&redlink=1" class="new" title="奇异性（页面不存在）">奇异性</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">在扩展到<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>的<a href="/wiki/%E8%A4%87%E5%B0%8D%E6%95%B8" title="複對數">复对数</a>情况下不能是1的<a href="/wiki/%E6%96%B9%E6%A0%B9" title="方根">方根</a></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">比尔吉受到了施蒂费尔相关工作的影响，他对等差数列和等比数列的关系作出了进一步的研究并于1610年前后发明了对数，但直到10年后（1620年），他才在《等差数列和等比数列表》中对外发布了他的思想。</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">对每个基<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 \alpha =|R|\neq 0,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =|R|\neq 0,1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1703d66806e91fc6b00912ee3bbace9a58f42b6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.101ex; height:2.843ex;" alt="{\displaystyle \alpha =|R|\neq 0,1}"></span>的值（不得是负数、0或1）只有唯一的对数函数。从这个角度看，底数<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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>的对数函数是<a href="/wiki/%E6%8C%87%E6%95%B0%E5%87%BD%E6%95%B0" title="指数函数">指数函数</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 y=\alpha ^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\alpha ^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797f11565df51770a27304ee77c1e66ca6974949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.914ex; height:2.676ex;" alt="{\displaystyle y=\alpha ^{x}}"></span>的<a href="/wiki/%E5%8F%8D%E5%87%BD%E6%95%B8" title="反函數">反函数</a>。词语“对数”经常用来称呼对数函数自身和这个函数的1个特定值。</span> </li> </ol></div> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=21" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-上海交通大学数学科学学院-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-上海交通大学数学科学学院_3-0"><sup><b>1.0</b></sup></a> <a href="#cite_ref-上海交通大学数学科学学院_3-1"><sup><b>1.1</b></sup></a> <a href="#cite_ref-上海交通大学数学科学学院_3-2"><sup><b>1.2</b></sup></a> <a href="#cite_ref-上海交通大学数学科学学院_3-3"><sup><b>1.3</b></sup></a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170606130611/http://www.math.sjtu.edu.cn/course/gdds/logarithm.htm">对数(logarithm)</a>. 上海交通大学数学科学学院. <span class="reference-accessdate"> [<span class="nowrap">2017-04-10</span>]</span>. （<a rel="nofollow" class="external text" href="http://www.math.sjtu.edu.cn/course/gdds/logarithm.htm">原始内容</a>存档于2017-06-06）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.btitle=%E5%AF%B9%E6%95%B0%28logarithm%29&rft.genre=unknown&rft.pub=%E4%B8%8A%E6%B5%B7%E4%BA%A4%E9%80%9A%E5%A4%A7%E5%AD%A6%E6%95%B0%E5%AD%A6%E7%A7%91%E5%AD%A6%E5%AD%A6%E9%99%A2&rft_id=http%3A%2F%2Fwww.math.sjtu.edu.cn%2Fcourse%2Fgdds%2Flogarithm.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></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">Much of the history of logarithms is derived from <i>The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions</i>, by James Mills Peirce, University Professor of Mathematics in Harvard University, 1873.</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"><cite class="citation book">史仲文. <a rel="nofollow" class="external text" href="http://www.readers365.com/zhongguolishi/mydoc087.htm">第087卷清代科技史五、数学（一）西方数学的传入与国人的研究１.对数方法的介绍</a>. 中国全史百卷本. <span class="reference-accessdate"> [<span class="nowrap">2017-04-10</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20200408105328/http://www.readers365.com/zhongguolishi/mydoc087.htm">存档</a>于2020-04-08）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.atitle=%E7%AC%AC087%E5%8D%B7+%E6%B8%85%E4%BB%A3%E7%A7%91%E6%8A%80%E5%8F%B2+%E4%BA%94%E3%80%81%E6%95%B0%E5%AD%A6+%EF%BC%88%E4%B8%80%EF%BC%89%E8%A5%BF%E6%96%B9%E6%95%B0%E5%AD%A6%E7%9A%84%E4%BC%A0%E5%85%A5%E4%B8%8E%E5%9B%BD%E4%BA%BA%E7%9A%84%E7%A0%94%E7%A9%B6+%EF%BC%91.%E5%AF%B9%E6%95%B0%E6%96%B9%E6%B3%95%E7%9A%84%E4%BB%8B%E7%BB%8D&rft.au=%E5%8F%B2%E4%BB%B2%E6%96%87&rft.btitle=%E4%B8%AD%E5%9B%BD%E5%85%A8%E5%8F%B2&rft.edition=%E7%99%BE%E5%8D%B7%E6%9C%AC&rft.genre=bookitem&rft_id=http%3A%2F%2Fwww.readers365.com%2Fzhongguolishi%2Fmydoc087.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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"><cite id="CITEREFDowning2003" class="citation">Downing, Douglas, Algebra the Easy Way, Barron's Educational Series, Hauppauge, N.Y.: Barron's, 2003, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-7641-1972-9" title="Special:网络书源/978-0-7641-1972-9"><span title="国际标准书号">ISBN</span> 978-0-7641-1972-9</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.aufirst=Douglas&rft.aulast=Downing&rft.btitle=Algebra+the+Easy+Way&rft.date=2003&rft.genre=book&rft.isbn=978-0-7641-1972-9&rft.place=Hauppauge%2C+N.Y.&rft.pub=Barron%27s&rft.series=Barron%27s+Educational+Series&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>, chapter 17, p. 275</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"><cite id="CITEREFWegener2005" class="citation">Wegener, Ingo, Complexity theory: exploring the limits of efficient algorithms, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, 2005, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-3-540-21045-0" title="Special:网络书源/978-3-540-21045-0"><span title="国际标准书号">ISBN</span> 978-3-540-21045-0</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.aufirst=Ingo&rft.aulast=Wegener&rft.btitle=Complexity+theory%3A+exploring+the+limits+of+efficient+algorithms&rft.date=2005&rft.genre=book&rft.isbn=978-3-540-21045-0&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>, p. 20</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"><cite id="CITEREFB._N._Taylor1995" class="citation">B. N. Taylor, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070629210131/http://physics.nist.gov/Pubs/SP811/sec10.html#10.1.2">Guide for the Use of the International System of Units (SI)</a>, US Department of Commerce, 1995 <span class="reference-accessdate"> [<span class="nowrap">2013-03-10</span>]</span>, （<a rel="nofollow" class="external text" href="http://physics.nist.gov/Pubs/SP811/sec10.html#10.1.2">原始内容</a>存档于2007-06-29）</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.au=B.+N.+Taylor&rft.btitle=Guide+for+the+Use+of+the+International+System+of+Units+%28SI%29&rft.date=1995&rft.genre=book&rft.pub=US+Department+of+Commerce&rft_id=http%3A%2F%2Fphysics.nist.gov%2FPubs%2FSP811%2Fsec10.html%2310.1.2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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"><cite id="CITEREFGullberg,_Jan1997" class="citation">Gullberg, Jan, Mathematics: from the birth of numbers., New York: W. W. Norton & Co, 1997, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-393-04002-9" title="Special:网络书源/978-0-393-04002-9"><span title="国际标准书号">ISBN</span> 978-0-393-04002-9</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.au=Gullberg%2C+Jan&rft.btitle=Mathematics%3A+from+the+birth+of+numbers.&rft.date=1997&rft.genre=book&rft.isbn=978-0-393-04002-9&rft.place=New+York&rft.pub=W.+W.+Norton+%26+Co&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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"> <cite id="CITEREFPaul_Halmos1985" class="citation">Paul Halmos, I Want to Be a Mathematician: An Automathography, Berlin, New York: Springer-Verlag, 1985, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-387-96078-4" title="Special:网络书源/978-0-387-96078-4"><span title="国际标准书号">ISBN</span> 978-0-387-96078-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.au=Paul+Halmos&rft.btitle=I+Want+to+Be+a+Mathematician%3A+An+Automathography&rft.date=1985&rft.genre=book&rft.isbn=978-0-387-96078-4&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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"> <cite id="CITEREFIrving_Stringham1893" class="citation">Irving Stringham, <a rel="nofollow" class="external text" href="http://books.google.com/?id=hPEKAQAAIAAJ&pg=PR13&dq=%22Irving+Stringham%22+In-natural-logarithm&q=">Uniplanar algebra: being part I of a propædeutic to the higher mathematical analysis</a>, The Berkeley Press: xiii, 1893</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.au=Irving+Stringham&rft.btitle=Uniplanar+algebra%3A+being+part+I+of+a+prop%C3%A6deutic+to+the+higher+mathematical+analysis&rft.date=1893&rft.genre=book&rft.pages=xiii&rft.pub=The+Berkeley+Press&rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DhPEKAQAAIAAJ%26pg%3DPR13%26dq%3D%2522Irving%2BStringham%2522%2BIn-natural-logarithm%26q%3D&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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"> <cite id="CITEREFRoy_S._Freedman2006" class="citation">Roy S. Freedman, <a rel="nofollow" class="external text" href="http://books.google.com/?id=APJ7QeR_XPkC&pg=PA59&dq=%22Irving+Stringham%22+logarithm+ln&q=%22Irving%20Stringham%22%20logarithm%20ln">Introduction to Financial Technology</a>, Amsterdam: Academic Press: 59, 2006, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-12-370478-8" title="Special:网络书源/978-0-12-370478-8"><span title="国际标准书号">ISBN</span> 978-0-12-370478-8</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%AF%B9%E6%95%B0&rft.au=Roy+S.+Freedman&rft.btitle=Introduction+to+Financial+Technology&rft.date=2006&rft.genre=book&rft.isbn=978-0-12-370478-8&rft.pages=59&rft.place=Amsterdam&rft.pub=Academic+Press&rft_id=http%3A%2F%2Fbooks.google.com%2F%3Fid%3DAPJ7QeR_XPkC%26pg%3DPA59%26dq%3D%2522Irving%2BStringham%2522%2Blogarithm%2Bln%26q%3D%2522Irving%2520Stringham%2522%2520logarithm%2520ln&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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"><i>Handbook of Mathematical Functions</i>, National Bureau of Standards (Applied Mathematics Series no.55), June 1964, page 68.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%AF%B9%E6%95%B0&action=edit&section=22" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.mathlogarithms.com/">Explaining Logarithms</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20080423111204/http://www.mathlogarithms.com/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070614192942/http://wolf.galekus.com/viewpage.php?page_id=10">Log Calculator for all bases.</a></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Logarithm.html">Logarithm</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20080430194051/http://mathworld.wolfram.com/Logarithm.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） on <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></li> <li><a rel="nofollow" class="external text" href="http://www.micheloud.com/FXM/LOG/index.htm">Jost Burgi, Swiss Inventor of Logarithms</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20080304115529/http://www.micheloud.com/FXM/LOG/index.htm">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="http://www.algebra.com/algebra/homework/logarithm/">Logarithm calculators and word problems with work shown, for school students</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20070927005900/http://www.algebra.com/algebra/homework/logarithm/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070627125949/http://www.johnnapier.com/table_of_logarithms_001.htm">Translation of Napier's work on logarithms</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070705203019/http://www.tufts.edu/%7Egdallal/logs.htm">Logarithms - from The Little Handbook of Statistical Practice</a></li> <li><a href="/w/index.php?title=Literateprograms:Logarithm_Function_(Python)&action=edit&redlink=1" class="new" title="Literateprograms:Logarithm Function (Python)（页面不存在）">Algorithm for determining Log values for any base</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080110163119/http://billeccentrec.blogspot.com/2007/07/tables-of-logarithms-and-trigonometric.html">常用對數表（文字版）</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84265675"><style data-mw-deduplicate="TemplateStyles:r84261037">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output 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