{"id":183491,"date":"2025-01-16T13:44:29","date_gmt":"2025-01-16T13:44:29","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=183491"},"modified":"2025-01-16T13:44:31","modified_gmt":"2025-01-16T13:44:31","slug":"solve-the-equation-for-x-log1-3-log1-2-0-5-0-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/16\/solve-the-equation-for-x-log1-3-log1-2-0-5-0-2\/","title":{"rendered":"Solve the equation (for x): log1-3-log1-2-0.5 = 0. 2"},"content":{"rendered":"\n

Solve the equation (for x): log1-3-log1-2-0.5 = 0. 2. Solve the equation (for x): log3 (38) 2-x. 3. List the first five terms of the sequence a\u2081 = 3, an+1 an n nEN.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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

Let’s break each part of the problem into steps to solve:<\/p>\n\n\n\n

\n\n\n\n

1. Solve the equation:<\/strong><\/h3>\n\n\n\n

[
\\log_{1-3} – \\log_{1-2} – 0.5 = 0
]
This equation is unclear because the logarithmic base ((\\log_{1-b})) is unconventional and may not follow typical logarithmic rules since (1-b \\leq 0) for most (b). Logarithms are only defined for positive bases not equal to 1. Therefore, this equation cannot be solved as written. Please clarify the equation (perhaps (\\log_{a}(b)) or (\\ln)) for further explanation.<\/p>\n\n\n\n

\n\n\n\n

2. Solve the equation:<\/strong><\/h3>\n\n\n\n

[
\\log_3(38) = 2 – x
]
Rewriting this equation:
[
x = 2 – \\log_3(38)
]
Using the change of base formula ((\\log_a(b) = \\frac{\\ln(b)}{\\ln(a)})):
[
\\log_3(38) = \\frac{\\ln(38)}{\\ln(3)}
]
Thus,
[
x = 2 – \\frac{\\ln(38)}{\\ln(3)}
]
Numerical approximation:
[
\\ln(38) \\approx 3.6376, \\quad \\ln(3) \\approx 1.0986
]
[
x = 2 – \\frac{3.6376}{1.0986} \\approx 2 – 3.31 = -1.31
]<\/p>\n\n\n\n

\n\n\n\n

3. List the first five terms of the sequence<\/strong><\/h3>\n\n\n\n

The sequence is defined as:
[
a_1 = 3, \\quad a_{n+1} = a_n \\cdot n
]<\/p>\n\n\n\n

First five terms:<\/h4>\n\n\n\n
    \n
  1. (a_1 = 3)<\/li>\n\n\n\n
  2. (a_2 = a_1 \\cdot 1 = 3 \\cdot 1 = 3)<\/li>\n\n\n\n
  3. (a_3 = a_2 \\cdot 2 = 3 \\cdot 2 = 6)<\/li>\n\n\n\n
  4. (a_4 = a_3 \\cdot 3 = 6 \\cdot 3 = 18)<\/li>\n\n\n\n
  5. (a_5 = a_4 \\cdot 4 = 18 \\cdot 4 = 72)<\/li>\n<\/ol>\n\n\n\n

    The first five terms are:
    [
    3, 3, 6, 18, 72
    ]<\/p>\n\n\n\n

    \n\n\n\n

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

    The above problems focus on solving logarithmic equations and understanding recursive sequences.<\/p>\n\n\n\n

    For the second equation, logarithms help us express exponential relationships. Using the change of base formula, we convert (\\log_3(38)) into a ratio of natural logarithms. After computing this value, we substitute it back to solve for (x). The change of base formula is a powerful tool when solving logarithmic equations involving uncommon bases.<\/p>\n\n\n\n

    For the sequence problem, we use a recursive formula where each term depends on the previous term and the index (n). Starting with the initial value (a_1 = 3), we calculate successive terms by multiplying the prior term by (n). Recursive sequences are widely used in mathematical modeling and algorithms.<\/p>\n\n\n\n

    If there are any clarifications or additional details required for part 1, please provide further information!<\/p>\n\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    Solve the equation (for x): log1-3-log1-2-0.5 = 0. 2. Solve the equation (for x): log3 (38) 2-x. 3. List the first five terms of the sequence a\u2081 = 3, an+1 an n nEN. The Correct Answer and Explanation is : Let’s break each part of the problem into steps to solve: 1. Solve the equation: […]<\/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-183491","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183491","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=183491"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183491\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183491"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183491"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}