Find the value of X in the exponential equation. 8^(2x-6) = 16^(2x-3).x = 2. Find the value of X in the exponential equation 1\/16 = 8^(x-3). x = (Answer should be in a fraction form, i.e 1\/2)<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n Let’s solve both exponential equations step by step.<\/p>\n\n\n\n [ Since: [ Since the bases are the same, we equate the exponents: [ [ [ We rewrite ( \\frac{1}{16} ) and ( 8 ) using base 2: [ Since the bases are the same, we equate the exponents: [ [ In both problems, we solve for (x) in an exponential equation by expressing the given bases as powers of a common base, typically 2, and then equating the exponents.<\/p>\n\n\n\n For the first equation, we rewrite (8) and (16) as powers of 2, leading to an equation where exponents are equal. Solving this linear equation gives (x = -3).<\/p>\n\n\n\n For the second equation, rewriting (1\/16) and (8) as base-2 expressions allows us to set the exponents equal, leading to another linear equation. Solving it results in (x = \\frac{5}{3}).<\/p>\n\n\n\n These techniques are useful in algebra and calculus for solving exponential and logarithmic equations efficiently.<\/p>\n","protected":false},"excerpt":{"rendered":" Find the value of X in the exponential equation. 8^(2x-6) = 16^(2x-3).x = 2. Find the value of X in the exponential equation 1\/16 = 8^(x-3). x = (Answer should be in a fraction form, i.e 1\/2) The Correct Answer and Explanation is : Let’s solve both exponential equations step by step. Problem 1: [8^{(2x-6)} […]<\/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-187702","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187702","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=187702"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/187702\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=187702"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=187702"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=187702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Problem 1:<\/strong><\/h3>\n\n\n\n
8^{(2x-6)} = 16^{(2x-3)}
]<\/p>\n\n\n\nStep 1: Express the bases as powers of 2<\/strong><\/h4>\n\n\n\n
[
8 = 2^3 \\quad \\text{and} \\quad 16 = 2^4
]
We rewrite the equation as:
[
(2^3)^{(2x-6)} = (2^4)^{(2x-3)}
]<\/p>\n\n\n\nStep 2: Apply the Power Rule ((a^m)^n = a^{m \\cdot n})<\/strong><\/h4>\n\n\n\n
2^{3(2x-6)} = 2^{4(2x-3)}
]
[
2^{(6x-18)} = 2^{(8x-12)}
]<\/p>\n\n\n\nStep 3: Set Exponents Equal to Each Other<\/strong><\/h4>\n\n\n\n
[
6x – 18 = 8x – 12
]<\/p>\n\n\n\nStep 4: Solve for (x)<\/strong><\/h4>\n\n\n\n
-18 + 12 = 8x – 6x
]
[
-6 = 2x
]
[
x = -\\frac{6}{2} = -3
]<\/p>\n\n\n\nAnswer for Problem 1:<\/strong><\/h3>\n\n\n\n
x = -3
]<\/p>\n\n\n\n
\n\n\n\nProblem 2:<\/strong><\/h3>\n\n\n\n
\\frac{1}{16} = 8^{(x-3)}
]<\/p>\n\n\n\nStep 1: Express the Bases as Powers of 2<\/strong><\/h4>\n\n\n\n
[
\\frac{1}{16} = 2^{-4} \\quad \\text{and} \\quad 8 = 2^3
]
Thus, we rewrite the equation as:
[
2^{-4} = (2^3)^{(x-3)}
]<\/p>\n\n\n\nStep 2: Apply the Power Rule ((a^m)^n = a^{m \\cdot n})<\/strong><\/h4>\n\n\n\n
2^{-4} = 2^{3(x-3)}
]<\/p>\n\n\n\nStep 3: Set Exponents Equal to Each Other<\/strong><\/h4>\n\n\n\n
[
-4 = 3(x – 3)
]<\/p>\n\n\n\nStep 4: Solve for (x)<\/strong><\/h4>\n\n\n\n
-4 = 3x – 9
]
[
-4 + 9 = 3x
]
[
5 = 3x
]
[
x = \\frac{5}{3}
]<\/p>\n\n\n\nAnswer for Problem 2:<\/strong><\/h3>\n\n\n\n
x = \\frac{5}{3}
]<\/p>\n\n\n\n
\n\n\n\nExplanation (300 words)<\/strong><\/h3>\n\n\n\n