We are given the logarithmic equation: log\u20614(x2\u22129)\u2212log\u20614(x+3)=log\u2061464\\log_4 (x^2 – 9) – \\log_4 (x + 3) = \\log_4 64<\/p>\n\n\n\n
\n\n\n\n
Step 1: Apply Logarithmic Rules<\/strong><\/h3>\n\n\n\nUsing the logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right)<\/p>\n\n\n\n
We simplify the left-hand side: log\u20614(x2\u22129x+3)=log\u2061464\\log_4 \\left( \\frac{x^2 – 9}{x + 3} \\right) = \\log_4 64<\/p>\n\n\n\n
\n\n\n\nStep 2: Eliminate the Logarithms<\/strong><\/h3>\n\n\n\nSince both sides have the same logarithmic base, we can equate the arguments: x2\u22129x+3=64\\frac{x^2 – 9}{x + 3} = 64<\/p>\n\n\n\n
\n\n\n\nStep 3: Simplify the Left Side<\/strong><\/h3>\n\n\n\nNotice that x2\u22129x^2 – 9 is a difference of squares: x2\u22129=(x\u22123)(x+3)x^2 – 9 = (x – 3)(x + 3)<\/p>\n\n\n\n
So, (x\u22123)(x+3)x+3\\frac{(x – 3)(x + 3)}{x + 3}<\/p>\n\n\n\n
As long as x\u2260\u22123x \\neq -3, the x+3x + 3 terms cancel: x\u22123=64x – 3 = 64<\/p>\n\n\n\n
\n\n\n\nStep 4: Solve for xx<\/strong><\/h3>\n\n\n\nx=64+3=67x = 64 + 3 = 67<\/p>\n\n\n\n
\n\n\n\nStep 5: Check for Restrictions<\/strong><\/h3>\n\n\n\nWe must check that the values inside the logarithms are positive:<\/p>\n\n\n\n
\n- x2\u22129=672\u22129=4489\u22129=4480>0x^2 – 9 = 67^2 – 9 = 4489 – 9 = 4480 > 0<\/li>\n\n\n\n
- x+3=67+3=70>0x + 3 = 67 + 3 = 70 > 0<\/li>\n<\/ul>\n\n\n\n
So, x=67x = 67 is a valid solution<\/strong>.<\/p>\n\n\n\n
\n\n\n\n\u2705 Final Answer:<\/h3>\n\n\n\n
67\\boxed{67}<\/p>\n\n\n\n
\n\n\n\nExplanation <\/strong><\/h3>\n\n\n\nThis logarithmic equation tests your understanding of logarithmic properties and algebraic manipulation. The original equation contains a difference of two logarithms on the left and a single logarithm on the right. Because all the logarithms share the same base (base 4), we can simplify the expression by using a logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right)<\/p>\n\n\n\n
This rule allows us to combine the two logarithms into a single one: log\u20614(x2\u22129x+3)\\log_4 \\left(\\frac{x^2 – 9}{x + 3}\\right). Once we simplify the rational expression inside the log, we notice that the numerator is a difference of squares. Factoring it gives (x\u22123)(x+3)(x – 3)(x + 3). When we divide this by x+3x + 3, the x+3x + 3 terms cancel out (as long as x\u2260\u22123x \\neq -3, which would make the denominator zero), and we are left with x\u22123x – 3.<\/p>\n\n\n\n
Now, we have log\u20614(x\u22123)=log\u2061464\\log_4 (x – 3) = \\log_4 64, so we equate the arguments and solve x\u22123=64x – 3 = 64, leading to x=67x = 67.<\/p>\n\n\n\n
We must check that this value doesn’t make any part of the logarithm undefined (logarithms are only defined for positive arguments). Since all values involved are positive when x=67x = 67, the solution is valid.<\/p>\n\n\n\n
Thus, the correct and verified solution is 67\\boxed{67}.<\/p>\n\n\n\n![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/05\/learnexams-banner9.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"NoneSolve: log4 (x2 – 9) – log4 (x + 3) = log4 64. The Correct Answer and Explanation is: We are given the logarithmic equation: log\u20614(x2\u22129)\u2212log\u20614(x+3)=log\u2061464\\log_4 (x^2 – 9) – \\log_4 (x + 3) = \\log_4 64 Step 1: Apply Logarithmic Rules Using the logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right) We […]<\/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-219066","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219066","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=219066"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219066\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219066"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219066"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219066"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Since both sides have the same logarithmic base, we can equate the arguments: x2\u22129x+3=64\\frac{x^2 – 9}{x + 3} = 64<\/p>\n\n\n\n
\n\n\n\n
Step 3: Simplify the Left Side<\/strong><\/h3>\n\n\n\nNotice that x2\u22129x^2 – 9 is a difference of squares: x2\u22129=(x\u22123)(x+3)x^2 – 9 = (x – 3)(x + 3)<\/p>\n\n\n\n
So, (x\u22123)(x+3)x+3\\frac{(x – 3)(x + 3)}{x + 3}<\/p>\n\n\n\n
As long as x\u2260\u22123x \\neq -3, the x+3x + 3 terms cancel: x\u22123=64x – 3 = 64<\/p>\n\n\n\n
\n\n\n\nStep 4: Solve for xx<\/strong><\/h3>\n\n\n\nx=64+3=67x = 64 + 3 = 67<\/p>\n\n\n\n
\n\n\n\nStep 5: Check for Restrictions<\/strong><\/h3>\n\n\n\nWe must check that the values inside the logarithms are positive:<\/p>\n\n\n\n
\n- x2\u22129=672\u22129=4489\u22129=4480>0x^2 – 9 = 67^2 – 9 = 4489 – 9 = 4480 > 0<\/li>\n\n\n\n
- x+3=67+3=70>0x + 3 = 67 + 3 = 70 > 0<\/li>\n<\/ul>\n\n\n\n
So, x=67x = 67 is a valid solution<\/strong>.<\/p>\n\n\n\n
\n\n\n\n\u2705 Final Answer:<\/h3>\n\n\n\n
67\\boxed{67}<\/p>\n\n\n\n
\n\n\n\nExplanation <\/strong><\/h3>\n\n\n\nThis logarithmic equation tests your understanding of logarithmic properties and algebraic manipulation. The original equation contains a difference of two logarithms on the left and a single logarithm on the right. Because all the logarithms share the same base (base 4), we can simplify the expression by using a logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right)<\/p>\n\n\n\n
This rule allows us to combine the two logarithms into a single one: log\u20614(x2\u22129x+3)\\log_4 \\left(\\frac{x^2 – 9}{x + 3}\\right). Once we simplify the rational expression inside the log, we notice that the numerator is a difference of squares. Factoring it gives (x\u22123)(x+3)(x – 3)(x + 3). When we divide this by x+3x + 3, the x+3x + 3 terms cancel out (as long as x\u2260\u22123x \\neq -3, which would make the denominator zero), and we are left with x\u22123x – 3.<\/p>\n\n\n\n
Now, we have log\u20614(x\u22123)=log\u2061464\\log_4 (x – 3) = \\log_4 64, so we equate the arguments and solve x\u22123=64x – 3 = 64, leading to x=67x = 67.<\/p>\n\n\n\n
We must check that this value doesn’t make any part of the logarithm undefined (logarithms are only defined for positive arguments). Since all values involved are positive when x=67x = 67, the solution is valid.<\/p>\n\n\n\n
Thus, the correct and verified solution is 67\\boxed{67}.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"NoneSolve: log4 (x2 – 9) – log4 (x + 3) = log4 64. The Correct Answer and Explanation is: We are given the logarithmic equation: log\u20614(x2\u22129)\u2212log\u20614(x+3)=log\u2061464\\log_4 (x^2 – 9) – \\log_4 (x + 3) = \\log_4 64 Step 1: Apply Logarithmic Rules Using the logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right) We […]<\/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-219066","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219066","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=219066"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219066\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219066"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219066"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219066"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
x=64+3=67x = 64 + 3 = 67<\/p>\n\n\n\n
\n\n\n\n
Step 5: Check for Restrictions<\/strong><\/h3>\n\n\n\nWe must check that the values inside the logarithms are positive:<\/p>\n\n\n\n
\n- x2\u22129=672\u22129=4489\u22129=4480>0x^2 – 9 = 67^2 – 9 = 4489 – 9 = 4480 > 0<\/li>\n\n\n\n
- x+3=67+3=70>0x + 3 = 67 + 3 = 70 > 0<\/li>\n<\/ul>\n\n\n\n
So, x=67x = 67 is a valid solution<\/strong>.<\/p>\n\n\n\n
\n\n\n\n\u2705 Final Answer:<\/h3>\n\n\n\n
67\\boxed{67}<\/p>\n\n\n\n
\n\n\n\nExplanation <\/strong><\/h3>\n\n\n\nThis logarithmic equation tests your understanding of logarithmic properties and algebraic manipulation. The original equation contains a difference of two logarithms on the left and a single logarithm on the right. Because all the logarithms share the same base (base 4), we can simplify the expression by using a logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right)<\/p>\n\n\n\n
This rule allows us to combine the two logarithms into a single one: log\u20614(x2\u22129x+3)\\log_4 \\left(\\frac{x^2 – 9}{x + 3}\\right). Once we simplify the rational expression inside the log, we notice that the numerator is a difference of squares. Factoring it gives (x\u22123)(x+3)(x – 3)(x + 3). When we divide this by x+3x + 3, the x+3x + 3 terms cancel out (as long as x\u2260\u22123x \\neq -3, which would make the denominator zero), and we are left with x\u22123x – 3.<\/p>\n\n\n\n
Now, we have log\u20614(x\u22123)=log\u2061464\\log_4 (x – 3) = \\log_4 64, so we equate the arguments and solve x\u22123=64x – 3 = 64, leading to x=67x = 67.<\/p>\n\n\n\n
We must check that this value doesn’t make any part of the logarithm undefined (logarithms are only defined for positive arguments). Since all values involved are positive when x=67x = 67, the solution is valid.<\/p>\n\n\n\n
Thus, the correct and verified solution is 67\\boxed{67}.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"NoneSolve: log4 (x2 – 9) – log4 (x + 3) = log4 64. The Correct Answer and Explanation is: We are given the logarithmic equation: log\u20614(x2\u22129)\u2212log\u20614(x+3)=log\u2061464\\log_4 (x^2 – 9) – \\log_4 (x + 3) = \\log_4 64 Step 1: Apply Logarithmic Rules Using the logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right) We […]<\/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-219066","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219066","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=219066"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219066\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219066"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219066"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219066"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
So, x=67x = 67 is a valid solution<\/strong>.<\/p>\n\n\n\n 67\\boxed{67}<\/p>\n\n\n\n This logarithmic equation tests your understanding of logarithmic properties and algebraic manipulation. The original equation contains a difference of two logarithms on the left and a single logarithm on the right. Because all the logarithms share the same base (base 4), we can simplify the expression by using a logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right)<\/p>\n\n\n\n This rule allows us to combine the two logarithms into a single one: log\u20614(x2\u22129x+3)\\log_4 \\left(\\frac{x^2 – 9}{x + 3}\\right). Once we simplify the rational expression inside the log, we notice that the numerator is a difference of squares. Factoring it gives (x\u22123)(x+3)(x – 3)(x + 3). When we divide this by x+3x + 3, the x+3x + 3 terms cancel out (as long as x\u2260\u22123x \\neq -3, which would make the denominator zero), and we are left with x\u22123x – 3.<\/p>\n\n\n\n Now, we have log\u20614(x\u22123)=log\u2061464\\log_4 (x – 3) = \\log_4 64, so we equate the arguments and solve x\u22123=64x – 3 = 64, leading to x=67x = 67.<\/p>\n\n\n\n We must check that this value doesn’t make any part of the logarithm undefined (logarithms are only defined for positive arguments). Since all values involved are positive when x=67x = 67, the solution is valid.<\/p>\n\n\n\n Thus, the correct and verified solution is 67\\boxed{67}.<\/p>\n\n\n\n NoneSolve: log4 (x2 – 9) – log4 (x + 3) = log4 64. The Correct Answer and Explanation is: We are given the logarithmic equation: log\u20614(x2\u22129)\u2212log\u20614(x+3)=log\u2061464\\log_4 (x^2 – 9) – \\log_4 (x + 3) = \\log_4 64 Step 1: Apply Logarithmic Rules Using the logarithmic identity: log\u2061bA\u2212log\u2061bB=log\u2061b(AB)\\log_b A – \\log_b B = \\log_b \\left(\\frac{A}{B}\\right) We […]<\/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-219066","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219066","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=219066"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/219066\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=219066"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=219066"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=219066"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\n\u2705 Final Answer:<\/h3>\n\n\n\n
\n\n\n\nExplanation <\/strong><\/h3>\n\n\n\n
<\/figure>\n","protected":false},"excerpt":{"rendered":"