{"id":225400,"date":"2025-06-04T07:58:41","date_gmt":"2025-06-04T07:58:41","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=225400"},"modified":"2025-06-04T07:58:43","modified_gmt":"2025-06-04T07:58:43","slug":"identify-the-solution-set-of-6-ln-e-eln-2x-2-3-6","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/04\/identify-the-solution-set-of-6-ln-e-eln-2x-2-3-6\/","title":{"rendered":"Identify the solution set of 6 ln e = eln 2x {2} {3} {6}"},"content":{"rendered":"\n
Identify the solution set of 6 ln e = eln 2x {2} {3} {6}<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
To solve the equation: 6ln\u2061e=eln\u20612×6 \\ln e = e^{\\ln 2x}<\/p>\n\n\n\n
\n\n\n\n
Step 1: Simplify the left side<\/strong><\/h3>\n\n\n\n
We use the identity: ln\u2061e=1\\ln e = 1<\/p>\n\n\n\n
So, 6ln\u2061e=6\u22c51=66 \\ln e = 6 \\cdot 1 = 6<\/p>\n\n\n\n
\n\n\n\n
Step 2: Simplify the right side<\/strong><\/h3>\n\n\n\n
Use the identity: eln\u2061a=a, for a>0e^{\\ln a} = a, \\text{ for } a > 0<\/p>\n\n\n\n
So: eln\u20612x=2xe^{\\ln 2x} = 2x<\/p>\n\n\n\n
\n\n\n\n
Step 3: Solve the equation<\/strong><\/h3>\n\n\n\n
Since we used ln\u20612x\\ln 2x, and logarithms are only defined for positive arguments, we must have: 2x>0\u21d2x>02x > 0 \\Rightarrow x > 0<\/p>\n\n\n\n
Our solution x=3x = 3 satisfies this condition.<\/p>\n\n\n\n
\n\n\n\n
\u2705 Final Answer:<\/h3>\n\n\n\n
The solution set is {3}\\{3\\}<\/strong>.<\/p>\n\n\n\n\n\n\n\n
\ud83d\udcd8 Explanation <\/h3>\n\n\n\n
The given equation is 6ln\u2061e=eln\u20612×6 \\ln e = e^{\\ln 2x}. To find the solution, we simplify both sides using logarithmic and exponential identities. The natural logarithm of ee, or ln\u2061e\\ln e, equals 1, so the left-hand side simplifies to 66.<\/p>\n\n\n\n
On the right-hand side, we see eln\u20612xe^{\\ln 2x}, which seems complicated at first glance. However, we apply a useful identity from logarithmic and exponential functions: for any positive real number aa, eln\u2061a=ae^{\\ln a} = a. This is because exponentiation and logarithms are inverse operations.<\/p>\n\n\n\n
Applying this identity, eln\u20612x=2xe^{\\ln 2x} = 2x, as long as<\/strong> 2x>02x > 0, meaning x>0x > 0. This is important because the logarithm is only defined for positive arguments, so 2x2x must be positive.<\/p>\n\n\n\n
So now, our simplified equation is: 6=2×6 = 2x<\/p>\n\n\n\n
To isolate xx, divide both sides by 2: x=3x = 3<\/p>\n\n\n\n
Finally, we confirm that this value satisfies all domain requirements \u2014 specifically, x=3>0x = 3 > 0, so 2x=6>02x = 6 > 0, and ln\u20612x\\ln 2x is defined.<\/p>\n\n\n\n
Thus, the only valid solution<\/strong> is x=3x = 3, and the solution set is {3}\\{3\\}<\/strong>. The other options ({2}, {6}) do not satisfy the equation.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"
Identify the solution set of 6 ln e = eln 2x {2} {3} {6} The Correct Answer and Explanation is: To solve the equation: 6ln\u2061e=eln\u20612×6 \\ln e = e^{\\ln 2x} Step 1: Simplify the left side We use the identity: ln\u2061e=1\\ln e = 1 So, 6ln\u2061e=6\u22c51=66 \\ln e = 6 \\cdot 1 = 6 Step […]<\/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-225400","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225400","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=225400"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225400\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=225400"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=225400"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=225400"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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