Solving Exponential and Logarithmic Equations In Exercise, solve for x.
500(1.075)120x = 100.000<\/p>\n\n\n\n
The correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To solve the equation 500(1.075)120x=100,000500(1.075)^{120x} = 100,000500(1.075)120x=100,000, we can follow these steps:<\/p>\n\n\n\n First, we need to isolate the term with the exponential expression. To do this, divide both sides of the equation by 500:(1.075)120x=100,000500(1.075)^{120x} = \\frac{100,000}{500}(1.075)120x=500100,000\u200b<\/p>\n\n\n\n Calculating the right side gives:(1.075)120x=200(1.075)^{120x} = 200(1.075)120x=200<\/p>\n\n\n\n Next, we apply the logarithm to both sides of the equation. You can use either natural logarithm (ln\u2061\\lnln) or common logarithm (log\u2061\\loglog). Here, we\u2019ll use natural logarithm:ln\u2061((1.075)120x)=ln\u2061(200)\\ln((1.075)^{120x}) = \\ln(200)ln((1.075)120x)=ln(200)<\/p>\n\n\n\n Using the logarithmic identity ln\u2061(ab)=b\u22c5ln\u2061(a)\\ln(a^b) = b \\cdot \\ln(a)ln(ab)=b\u22c5ln(a), we can simplify the left side:120x\u22c5ln\u2061(1.075)=ln\u2061(200)120x \\cdot \\ln(1.075) = \\ln(200)120x\u22c5ln(1.075)=ln(200)<\/p>\n\n\n\n Now, we can isolate xxx by dividing both sides by 120\u22c5ln\u2061(1.075)120 \\cdot \\ln(1.075)120\u22c5ln(1.075):x=ln\u2061(200)120\u22c5ln\u2061(1.075)x = \\frac{\\ln(200)}{120 \\cdot \\ln(1.075)}x=120\u22c5ln(1.075)ln(200)\u200b<\/p>\n\n\n\n Next, we need to compute the values.<\/p>\n\n\n\n Thus, the solution for xxx is approximately:0.609\\boxed{0.609}0.609\u200b<\/p>\n\n\n\n This solution involves isolating the exponential expression and applying logarithms, which is a standard technique for solving equations involving exponents. The use of logarithms allows us to turn the exponentiation into a product, which is easier to manipulate. Calculating the natural logarithm provides specific numerical values that can be used to solve for xxx. This process is widely applicable in exponential growth or decay problems in various fields, including finance and science, making it an essential skill in mathematics.<\/p>\n","protected":false},"excerpt":{"rendered":" Solving Exponential and Logarithmic Equations In Exercise, solve for x.500(1.075)120x = 100.000 The correct Answer and Explanation is: To solve the equation 500(1.075)120x=100,000500(1.075)^{120x} = 100,000500(1.075)120x=100,000, we can follow these steps: Step 1: Isolate the Exponential Expression First, we need to isolate the term with the exponential expression. To do this, divide both sides of the […]<\/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-147540","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/147540","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=147540"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/147540\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=147540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=147540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=147540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Isolate the Exponential Expression<\/h3>\n\n\n\n
Step 2: Take the Logarithm of Both Sides<\/h3>\n\n\n\n
Step 3: Solve for xxx<\/h3>\n\n\n\n
Step 4: Calculate the Values<\/h3>\n\n\n\n
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Final Answer<\/h3>\n\n\n\n
Explanation<\/h3>\n\n\n\n