{"id":168888,"date":"2024-11-18T17:02:28","date_gmt":"2024-11-18T17:02:28","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=168888"},"modified":"2024-11-18T17:02:32","modified_gmt":"2024-11-18T17:02:32","slug":"what-is-the-solution-to-the-equation-below-log6-4x-2-log6-x2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/18\/what-is-the-solution-to-the-equation-below-log6-4x-2-log6-x2\/","title":{"rendered":"What is the solution to the equation below log6 4x 2 – log6 x=2"},"content":{"rendered":"\n

What is the solution to the equation below? log6 4x 2 – log6 x=2<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To solve the equation log\u20616(4×2)\u2212log\u20616(x)=2\\log_6 (4x^2) – \\log_6 (x) = 2log6\u200b(4×2)\u2212log6\u200b(x)=2, follow these steps:<\/p>\n\n\n\n

Step 1: Simplify the logarithmic expression<\/h3>\n\n\n\n

Using the logarithmic property<\/strong> log\u2061b(A)\u2212log\u2061b(B)=log\u2061b(AB)\\log_b (A) – \\log_b (B) = \\log_b \\left(\\frac{A}{B}\\right)logb\u200b(A)\u2212logb\u200b(B)=logb\u200b(BA\u200b), rewrite the left-hand side:log\u20616(4×2)\u2212log\u20616(x)=log\u20616(4x2x)\\log_6 (4x^2) – \\log_6 (x) = \\log_6 \\left(\\frac{4x^2}{x}\\right)log6\u200b(4×2)\u2212log6\u200b(x)=log6\u200b(x4x2\u200b)<\/p>\n\n\n\n

Simplify the fraction 4x2x\\frac{4x^2}{x}x4x2\u200b:log\u20616(4x2x)=log\u20616(4x)\\log_6 \\left(\\frac{4x^2}{x}\\right) = \\log_6 (4x)log6\u200b(x4x2\u200b)=log6\u200b(4x)<\/p>\n\n\n\n

The equation now becomes:log\u20616(4x)=2\\log_6 (4x) = 2log6\u200b(4x)=2<\/p>\n\n\n\n

\n\n\n\n

Step 2: Rewrite the logarithmic equation in exponential form<\/h3>\n\n\n\n

Using the definition of a logarithm log\u2061b(A)=C\u2005\u200a\u27fa\u2005\u200abC=A\\log_b (A) = C \\iff b^C = Alogb\u200b(A)=C\u27fabC=A, rewrite log\u20616(4x)=2\\log_6 (4x) = 2log6\u200b(4x)=2:62=4×6^2 = 4×62=4x<\/p>\n\n\n\n

Simplify 626^262:36=4×36 = 4×36=4x<\/p>\n\n\n\n

\n\n\n\n

Step 3: Solve for xxx<\/h3>\n\n\n\n

Divide both sides of the equation by 4:x=364=9x = \\frac{36}{4} = 9x=436\u200b=9<\/p>\n\n\n\n

\n\n\n\n

Step 4: Verify the solution<\/h3>\n\n\n\n

Substitute x=9x = 9x=9 back into the original equation to confirm:log\u20616(4(9)2)\u2212log\u20616(9)=log\u20616(4\u22c581)\u2212log\u20616(9)=log\u20616(324)\u2212log\u20616(9)\\log_6 (4(9)^2) – \\log_6 (9) = \\log_6 (4 \\cdot 81) – \\log_6 (9) = \\log_6 (324) – \\log_6 (9)log6\u200b(4(9)2)\u2212log6\u200b(9)=log6\u200b(4\u22c581)\u2212log6\u200b(9)=log6\u200b(324)\u2212log6\u200b(9)<\/p>\n\n\n\n

Using the logarithmic property:log\u20616(324)\u2212log\u20616(9)=log\u20616(3249)=log\u20616(36)\\log_6 (324) – \\log_6 (9) = \\log_6 \\left(\\frac{324}{9}\\right) = \\log_6 (36)log6\u200b(324)\u2212log6\u200b(9)=log6\u200b(9324\u200b)=log6\u200b(36)<\/p>\n\n\n\n

Since log\u20616(36)=2\\log_6 (36) = 2log6\u200b(36)=2, the solution is correct.<\/p>\n\n\n\n

\n\n\n\n

Final Answer:<\/h3>\n\n\n\n

x=9x = 9x=9<\/p>\n\n\n\n

\n\n\n\n

Explanation (300 words)<\/h3>\n\n\n\n

This problem involves logarithmic properties to simplify and solve an equation. The key idea is to apply the subtraction rule of logarithms, log\u2061b(A)\u2212log\u2061b(B)=log\u2061b(AB)\\log_b (A) – \\log_b (B) = \\log_b \\left(\\frac{A}{B}\\right)logb\u200b(A)\u2212logb\u200b(B)=logb\u200b(BA\u200b), which reduces two logarithms into one. Here, log\u20616(4×2)\u2212log\u20616(x)\\log_6 (4x^2) – \\log_6 (x)log6\u200b(4×2)\u2212log6\u200b(x) simplifies to log\u20616(4x2x)=log\u20616(4x)\\log_6 \\left(\\frac{4x^2}{x}\\right) = \\log_6 (4x)log6\u200b(x4x2\u200b)=log6\u200b(4x).<\/p>\n\n\n\n

After simplifying, the equation becomes log\u20616(4x)=2\\log_6 (4x) = 2log6\u200b(4x)=2. The next step is converting the logarithmic equation into its exponential form. By definition, log\u2061b(A)=C\\log_b (A) = Clogb\u200b(A)=C is equivalent to bC=Ab^C = AbC=A. Using this, log\u20616(4x)=2\\log_6 (4x) = 2log6\u200b(4x)=2 translates to 62=4×6^2 = 4×62=4x, or 36=4×36 = 4×36=4x. Solving for xxx gives x=9x = 9x=9.<\/p>\n\n\n\n

Verification confirms the solution: substituting x=9x = 9x=9 back into the original equation produces a true statement, proving the result is correct. This approach showcases how logarithmic rules and exponential transformations simplify complex equations.<\/p>\n","protected":false},"excerpt":{"rendered":"

What is the solution to the equation below? log6 4x 2 – log6 x=2 The Correct Answer and Explanation is: To solve the equation log\u20616(4×2)\u2212log\u20616(x)=2\\log_6 (4x^2) – \\log_6 (x) = 2log6\u200b(4×2)\u2212log6\u200b(x)=2, follow these steps: Step 1: Simplify the logarithmic expression Using the logarithmic property log\u2061b(A)\u2212log\u2061b(B)=log\u2061b(AB)\\log_b (A) – \\log_b (B) = \\log_b \\left(\\frac{A}{B}\\right)logb\u200b(A)\u2212logb\u200b(B)=logb\u200b(BA\u200b), rewrite the left-hand […]<\/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-168888","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/168888","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=168888"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/168888\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=168888"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=168888"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=168888"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}