{"id":205554,"date":"2025-03-25T05:25:33","date_gmt":"2025-03-25T05:25:33","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=205554"},"modified":"2025-03-25T05:25:35","modified_gmt":"2025-03-25T05:25:35","slug":"evaluate-log25-log-125-1-2-log-625-3-log-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/25\/evaluate-log25-log-125-1-2-log-625-3-log-5\/","title":{"rendered":"Evaluate: Log25-log 125 + 1\/2 log 625\/3 log 5"},"content":{"rendered":"\n<p>Evaluate: Log25-log 125 + 1\/2 log 625\/3 log 5<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>We will evaluate the given expression step by step:<\/p>\n\n\n\n<p>[<br>\\log 25 &#8211; \\log 125 + \\frac{1}{2} \\log \\frac{625}{3 \\log 5}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Express all terms in base 5<\/h3>\n\n\n\n<p>We know that:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( 25 = 5^2 \\Rightarrow \\log 25 = \\log 5^2 = 2 \\log 5 )<\/li>\n\n\n\n<li>( 125 = 5^3 \\Rightarrow \\log 125 = \\log 5^3 = 3 \\log 5 )<\/li>\n\n\n\n<li>( 625 = 5^4 \\Rightarrow \\log 625 = \\log 5^4 = 4 \\log 5 )<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Use logarithmic properties<\/h3>\n\n\n\n<p>The given expression simplifies as follows:<\/p>\n\n\n\n<p>[<br>2\\log 5 &#8211; 3\\log 5 + \\frac{1}{2} \\log \\frac{4\\log 5}{3\\log 5}<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Simplify each term<\/h3>\n\n\n\n<p>[<br>2\\log 5 &#8211; 3\\log 5 = -\\log 5<br>]<\/p>\n\n\n\n<p>Now, evaluate the fraction inside the logarithm:<br>[<br>\\frac{4\\log 5}{3\\log 5} = \\frac{4}{3}<br>]<\/p>\n\n\n\n<p>Taking the logarithm,<br>[<br>\\log \\frac{4}{3} = \\log 4 &#8211; \\log 3<br>]<\/p>\n\n\n\n<p>Since ( \\log 4 = 2\\log 2 ) and using approximate values,<br>[<br>\\log 2 \\approx 0.301 \\Rightarrow \\log 4 = 2(0.301) = 0.602<br>]<\/p>\n\n\n\n<p>Also,<br>[<br>\\log 3 \\approx 0.477<br>]<\/p>\n\n\n\n<p>So,<br>[<br>\\log \\frac{4}{3} = 0.602 &#8211; 0.477 = 0.125<br>]<\/p>\n\n\n\n<p>Now multiply by ( \\frac{1}{2} ),<br>[<br>\\frac{1}{2} \\times 0.125 = 0.0625<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Final Computation<\/h3>\n\n\n\n<p>[<br>-\\log 5 + 0.0625<br>]<\/p>\n\n\n\n<p>Approximating ( \\log 5 \\approx 0.699 ),<br>[<br>-0.699 + 0.0625 = -0.6365<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Answer:<\/h3>\n\n\n\n<p>[<br>\\mathbf{-0.6365}<br>]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation (300 Words)<\/h3>\n\n\n\n<p>Logarithms simplify calculations involving exponents by converting multiplication into addition and division into subtraction. In this problem, we simplify each term using logarithmic identities.<\/p>\n\n\n\n<p>First, we express 25, 125, and 625 in terms of base 5 because logarithm properties allow simplifications:<br>[<br>\\log 25 = 2 \\log 5, \\quad \\log 125 = 3 \\log 5, \\quad \\log 625 = 4 \\log 5<br>]<br>Using the logarithm subtraction rule:<br>[<br>\\log a &#8211; \\log b = \\log \\left( \\frac{a}{b} \\right)<br>]<br>We simplify ( 2 \\log 5 &#8211; 3 \\log 5 = -\\log 5 ).<\/p>\n\n\n\n<p>Next, we evaluate ( \\frac{1}{2} \\log \\frac{625}{3 \\log 5} ), which simplifies to ( \\frac{1}{2} \\log \\frac{4}{3} ).<br>Approximating logarithm values:<br>[<br>\\log \\frac{4}{3} = 0.125<br>]<br>Multiplying by ( \\frac{1}{2} ),<br>[<br>\\frac{1}{2} \\times 0.125 = 0.0625<br>]<\/p>\n\n\n\n<p>Finally, adding all terms gives the answer:<br>[<br>-0.6365<br>]<\/p>\n\n\n\n<p>Now, I&#8217;ll generate an image related to logarithmic functions.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-1343.png\" alt=\"\" class=\"wp-image-205555\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Evaluate: Log25-log 125 + 1\/2 log 625\/3 log 5 The correct answer and explanation is : We will evaluate the given expression step by step: [\\log 25 &#8211; \\log 125 + \\frac{1}{2} \\log \\frac{625}{3 \\log 5}] Step 1: Express all terms in base 5 We know that: Step 2: Use logarithmic properties The given expression [&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-205554","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/205554","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=205554"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/205554\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=205554"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=205554"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=205554"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}