{"id":14256,"date":"2023-05-01T10:00:11","date_gmt":"2023-05-01T04:30:11","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=14256"},"modified":"2023-05-02T10:14:19","modified_gmt":"2023-05-02T04:44:19","slug":"logarithms-work","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/logarithms-work\/","title":{"rendered":"How do Logarithms Work?"},"content":{"rendered":"<h2>How do <a href=\"https:\/\/www.aplustopper.com\/logarithmic-expressions\/\">Logarithms<\/a> Work?<\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/4UNkQcBrLaQ?start=6&#038;feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Logarithms<\/h3>\n<p>\u201cThe Logarithm of a given number to a given base is the index of the power to which the base must be raised in order to equal the given number.\u201d<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130084\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-1.jpg\" alt=\"How do Logarithms Work 1\" width=\"705\" height=\"430\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-1.jpg 705w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-1-300x183.jpg 300w\" sizes=\"(max-width: 705px) 100vw, 705px\" \/><br \/>\nIf a &gt; 0 and \u2260 1 then logarithm of a positive number N is defined as the index x of that power of &#8216;a&#8217; which equals N i.e.,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130085\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-2.jpg\" alt=\"How do Logarithms Work 2\" width=\"522\" height=\"34\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-2.jpg 522w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-2-300x20.jpg 300w\" sizes=\"(max-width: 522px) 100vw, 522px\" \/><br \/>\nIt is also known as fundamental logarithmic identity.<br \/>\nIts domain is (0, \u221e) and range is R. a is called the base of the logarithmic function.<\/p>\n<p>When base is &#8216;e&#8217; then the logarithmic function is called <strong>natural<\/strong> or <strong>Napierian<\/strong> <strong>logarithmic function<\/strong> and when base is 10, then it is called common logarithmic function.<\/p>\n<h3>Characteristic and mantissa<\/h3>\n<ol>\n<li>The integral part of a logarithm is called the characteristic and the fractional part is called mantissa.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130086\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-3.jpg\" alt=\"How do Logarithms Work 3\" width=\"350\" height=\"67\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-3.jpg 350w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-3-300x57.jpg 300w\" sizes=\"(max-width: 350px) 100vw, 350px\" \/><\/li>\n<li>The mantissa part of log of a number is always kept positive.<\/li>\n<li>If the characteristics of log<sub>10<\/sub> N\u00a0be n, then the number of digits in N is (n+1).<\/li>\n<li>If the characteristics of log<sub>10<\/sub> N\u00a0be (\u2013 n) then there exists (n \u2013 1) number of zeros after decimal part of N.<\/li>\n<\/ol>\n<h3>Properties of logarithms<\/h3>\n<p>Let <em>m<\/em> and <em>n<\/em> be arbitrary positive numbers such that a &gt; 0, a \u2260 1, b &gt; 0, b \u2260 1 then<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130088\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-4.jpg\" alt=\"How do Logarithms Work 4\" width=\"472\" height=\"469\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-4.jpg 472w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-4-150x150.jpg 150w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-4-300x298.jpg 300w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-4-100x100.jpg 100w\" sizes=\"(max-width: 472px) 100vw, 472px\" \/><\/p>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/YxrL-pIMws0?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Logarithmic inequalities<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130089\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-5.jpg\" alt=\"How do Logarithms Work 5\" width=\"379\" height=\"406\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-5.jpg 379w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/How-do-Logarithms-Work-5-280x300.jpg 280w\" sizes=\"(max-width: 379px) 100vw, 379px\" \/><br \/>\n(10) log <em><sub>p<\/sub><\/em> <em>a<\/em> &gt; log <em><sub>p<\/sub><\/em> <em>b<\/em> \u21d2 <em>a<\/em> \u2265 <em>b<\/em> if base <em>p<\/em> is positive and &gt;1 or <em>a<\/em> \u2264 <em>b<\/em> if base <em>p<\/em> is positive and &lt; 1 <em>i.e.,<\/em> 0 &lt; <em>p<\/em> &lt; 1.<br \/>\nIn other words, if base is greater than 1 then inequality remains same and if base is positive but less than 1 then the sign of inequality is reversed.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>How do Logarithms Work? Logarithms \u201cThe Logarithm of a given number to a given base is the index of the power to which the base must be raised in order to equal the given number.\u201d If a &gt; 0 and \u2260 1 then logarithm of a positive number N is defined as the index x [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[5465,5467,5464,5466],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>How do Logarithms Work? - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/logarithms-work\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"How do Logarithms Work?\" \/>\n<meta property=\"og:description\" content=\"How do Logarithms Work? 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