{"id":250524,"date":"2025-07-10T10:13:39","date_gmt":"2025-07-10T10:13:39","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=250524"},"modified":"2025-07-10T10:13:41","modified_gmt":"2025-07-10T10:13:41","slug":"evaluate-the-logarithmic-expression","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/10\/evaluate-the-logarithmic-expression\/","title":{"rendered":"Evaluate the logarithmic expression"},"content":{"rendered":"\n<p>Evaluate the logarithmic expression: log8(1\/16)<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To evaluate the logarithmic expression log\u20618(116)\\log_8 \\left(\\frac{1}{16}\\right)log8\u200b(161\u200b), we need to understand what the logarithm represents. The general form of a logarithmic expression is:log\u2061b(x)=yif&nbsp;and&nbsp;only&nbsp;ifby=x\\log_b(x) = y \\quad \\text{if and only if} \\quad b^y = xlogb\u200b(x)=yif&nbsp;and&nbsp;only&nbsp;ifby=x<\/p>\n\n\n\n<p>In this case, b=8b = 8b=8, and x=116x = \\frac{1}{16}x=161\u200b. So, we are looking for the exponent yyy such that:8y=1168^y = \\frac{1}{16}8y=161\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Express 8 and 16 as powers of 2<\/h3>\n\n\n\n<p>To make the calculation easier, we can express both 8 and 16 as powers of 2:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>8=238 = 2^38=23<\/li>\n\n\n\n<li>16=2416 = 2^416=24<\/li>\n<\/ul>\n\n\n\n<p>So, the expression becomes:8y=116=16\u22121=(24)\u22121=2\u221248^y = \\frac{1}{16} = 16^{-1} = (2^4)^{-1} = 2^{-4}8y=161\u200b=16\u22121=(24)\u22121=2\u22124<\/p>\n\n\n\n<p>Now the equation becomes:(23)y=2\u22124(2^3)^y = 2^{-4}(23)y=2\u22124<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Apply the power rule<\/h3>\n\n\n\n<p>Use the power of a power rule (am)n=am\u22c5n(a^m)^n = a^{m \\cdot n}(am)n=am\u22c5n to simplify the left-hand side:23y=2\u221242^{3y} = 2^{-4}23y=2\u22124<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Solve for yyy<\/h3>\n\n\n\n<p>Since the bases on both sides are the same, we can equate the exponents:3y=\u221243y = -43y=\u22124<\/p>\n\n\n\n<p>Now, solve for yyy:y=\u221243y = \\frac{-4}{3}y=3\u22124\u200b<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<p>Thus, the value of log\u20618(116)\\log_8 \\left(\\frac{1}{16}\\right)log8\u200b(161\u200b) is \u221243\\frac{-4}{3}3\u22124\u200b.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>In this process, we converted both the base (8) and the argument (16) to powers of 2, which allowed us to compare their exponents directly. The final solution shows that to raise 8 to the power of \u221243\\frac{-4}{3}3\u22124\u200b results in 116\\frac{1}{16}161\u200b.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-120.jpeg\" alt=\"\" class=\"wp-image-250525\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Evaluate the logarithmic expression: log8(1\/16) The Correct Answer and Explanation is: To evaluate the logarithmic expression log\u20618(116)\\log_8 \\left(\\frac{1}{16}\\right)log8\u200b(161\u200b), we need to understand what the logarithm represents. The general form of a logarithmic expression is:log\u2061b(x)=yif&nbsp;and&nbsp;only&nbsp;ifby=x\\log_b(x) = y \\quad \\text{if and only if} \\quad b^y = xlogb\u200b(x)=yif&nbsp;and&nbsp;only&nbsp;ifby=x In this case, b=8b = 8b=8, and x=116x = \\frac{1}{16}x=161\u200b. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-250524","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250524","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=250524"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/250524\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=250524"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=250524"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=250524"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}