x = -8 x = -4 x = 4 x = 8,<\/p>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/image-301.png\")
<\/figure>\n\n\n\nThe Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n We are given the equation: 3log\u20614x=log\u2061432+log\u2061423 \\log_4 x = \\log_4 32 + \\log_4 23log4\u200bx=log4\u200b32+log4\u200b2<\/p>\n\n\n\n The right-hand side uses the property of logarithms: log\u2061bA+log\u2061bB=log\u2061b(AB)\\log_b A + \\log_b B = \\log_b (AB)logb\u200bA+logb\u200bB=logb\u200b(AB)<\/p>\n\n\n\n So we combine the right-hand side: log\u2061432+log\u206142=log\u20614(32\u00d72)=log\u2061464\\log_4 32 + \\log_4 2 = \\log_4 (32 \\times 2) = \\log_4 64log4\u200b32+log4\u200b2=log4\u200b(32\u00d72)=log4\u200b64<\/p>\n\n\n\n Now the equation becomes: 3log\u20614x=log\u20614643 \\log_4 x = \\log_4 643log4\u200bx=log4\u200b64<\/p>\n\n\n\n Use the power property: alog\u2061bx=log\u2061b(xa)a \\log_b x = \\log_b (x^a)alogb\u200bx=logb\u200b(xa)<\/p>\n\n\n\n So: 3log\u20614x=log\u20614(x3)3 \\log_4 x = \\log_4 (x^3)3log4\u200bx=log4\u200b(x3)<\/p>\n\n\n\n Now the equation is: log\u20614(x3)=log\u2061464\\log_4 (x^3) = \\log_4 64log4\u200b(x3)=log4\u200b64<\/p>\n\n\n\n Since the bases and logs are equal, we equate the arguments: x3=64x^3 = 64×3=64<\/p>\n\n\n\n Take the cube root of both sides: x=643=4x = \\sqrt[3]{64} = 4x=364\u200b=4<\/p>\n\n\n\n x = 4<\/strong><\/p>\n\n\n\n This problem tests your understanding of logarithmic rules, particularly properties for combining and simplifying logarithmic expressions. The given equation includes a logarithm with a base of 4, and we need to simplify both sides to isolate the variable xxx.<\/p>\n\n\n\n First, we simplify the right-hand side using the product rule of logarithms: log\u2061bA+log\u2061bB=log\u2061b(AB)\\log_b A + \\log_b B = \\log_b (AB)logb\u200bA+logb\u200bB=logb\u200b(AB). Applying this, the sum log\u2061432+log\u206142\\log_4 32 + \\log_4 2log4\u200b32+log4\u200b2 becomes log\u20614(32\u22c52)=log\u2061464\\log_4 (32 \\cdot 2) = \\log_4 64log4\u200b(32\u22c52)=log4\u200b64. Recognizing powers of 2 helps here: 32=2532 = 2^532=25 and 2=212 = 2^12=21, so 64=2664 = 2^664=26, which is important for evaluating logarithms.<\/p>\n\n\n\n Next, on the left-hand side, we simplify 3log\u20614×3 \\log_4 x3log4\u200bx using the power rule of logarithms: alog\u2061bx=log\u2061b(xa)a \\log_b x = \\log_b (x^a)alogb\u200bx=logb\u200b(xa). Thus, 3log\u20614x=log\u20614(x3)3 \\log_4 x = \\log_4 (x^3)3log4\u200bx=log4\u200b(x3).<\/p>\n\n\n\n Now the equation becomes log\u20614(x3)=log\u2061464\\log_4 (x^3) = \\log_4 64log4\u200b(x3)=log4\u200b64. Because the logs have the same base and the expressions are equal, their arguments must be equal too. That gives us the equation x3=64x^3 = 64×3=64.<\/p>\n\n\n\n To solve x3=64x^3 = 64×3=64, take the cube root of both sides. Since 64=4364 = 4^364=43, we find x=4x = 4x=4.<\/p>\n\n\n\n In conclusion, using logarithmic properties allowed us to transform and simplify the equation, ultimately solving it by equating arguments of the logs. The correct solution is: What is the solution to the equation below? 3log4\u2061x=log4\u206132+log4\u20612 x = -8 x = -4 x = 4 x = 8x = -8 x = -4 x = 4 x = 8, The Correct Answer and Explanation is: We are given the equation: 3log\u20614x=log\u2061432+log\u2061423 \\log_4 x = \\log_4 32 + \\log_4 23log4\u200bx=log4\u200b32+log4\u200b2 Step-by-Step Solution Step 1: Use […]<\/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-229880","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229880","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=229880"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/229880\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=229880"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=229880"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=229880"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step-by-Step Solution<\/h3>\n\n\n\n
Step 1: Use the property of logarithms<\/h4>\n\n\n\n
Step 2: Use the property of logarithms<\/h4>\n\n\n\n
Step 3: Solve the equation<\/h4>\n\n\n\n
\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
\n\n\n\nExplanation<\/h3>\n\n\n\n
x = 4<\/strong>.<\/p>\n\n\n\n![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-144.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"