{"id":271992,"date":"2025-07-26T08:56:23","date_gmt":"2025-07-26T08:56:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=271992"},"modified":"2025-07-26T08:56:25","modified_gmt":"2025-07-26T08:56:25","slug":"directions-using-the-digits-1-to-9-fill-in-the-boxes-to-produce-log-that-meets-the-indicated-requirements","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/26\/directions-using-the-digits-1-to-9-fill-in-the-boxes-to-produce-log-that-meets-the-indicated-requirements\/","title":{"rendered":"Directions: Using the digits 1 to 9, fill in the boxes to produce log that meets the indicated requirements."},"content":{"rendered":"\n<p>Show transcrip<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/image-769.png\" alt=\"\" class=\"wp-image-271993\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Here is a correct solution to the puzzle.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>log\u2083(1 * 9)\u00a0Produces an integer<\/li>\n\n\n\n<li>log\u2085(6 \/ 7)\u00a0Produces an irrational number<\/li>\n\n\n\n<li>log\u2088(4\u00b2)\u00a0Produces a rational number<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>This puzzle requires using the digits 1 through 9 exactly once to satisfy three conditions involving logarithms. The solution is found by understanding the properties that make a logarithm an integer, rational, or irrational number.<\/p>\n\n\n\n<p>The most restrictive condition is producing a rational number from&nbsp;log_b(x^y). For a logarithm&nbsp;log_b(x)&nbsp;to be rational, its base&nbsp;b&nbsp;and its argument&nbsp;x&nbsp;must both be integer powers of the same root number. For example, the set {2, 4, 8} are all powers of 2, and {3, 9} are powers of 3. Using the property&nbsp;log_b(x^y) = y * log_b(x), we can select&nbsp;b=8&nbsp;and&nbsp;x=4. The value&nbsp;log\u2088(4)&nbsp;is 2\/3, because&nbsp;8^(2\/3) = (\u221b8)\u00b2 = 2\u00b2 = 4. By choosing&nbsp;y=2, the expression becomes&nbsp;log\u2088(4\u00b2), which simplifies to&nbsp;2 * log\u2088(4) = 2 * (2\/3) = 4\/3. This is a rational number. This step uses the digits 2, 4, and 8.<\/p>\n\n\n\n<p>Next, we address the integer condition,&nbsp;log_b(x * y). For a logarithm to be an integer&nbsp;k, the argument must be the base raised to the power of&nbsp;k, or&nbsp;b^k = x * y. From our remaining digits {1, 3, 5, 6, 7, 9}, we can choose&nbsp;b=3. If we let the integer result be&nbsp;k=2, then we need&nbsp;x * y = 3\u00b2 = 9. Using the digits 1 and 9 for&nbsp;x&nbsp;and&nbsp;y, the expression&nbsp;log\u2083(1 * 9)&nbsp;equals&nbsp;log\u2083(9), which is 2, a perfect integer. This uses the digits 1, 3, and 9.<\/p>\n\n\n\n<p>Finally, the remaining digits {5, 6, 7} are used for the irrational number condition,&nbsp;log_b(x \/ y). An irrational logarithm occurs when the base and argument do not share a common power root. Placing the digits gives us&nbsp;log\u2085(6 \/ 7). Since 5 and the fraction 6\/7 cannot be expressed as integer powers of the same base number, the result is guaranteed to be an irrational number.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner6-2116.jpeg\" alt=\"\" class=\"wp-image-271994\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Show transcrip The Correct Answer and Explanation is: Here is a correct solution to the puzzle. Explanation This puzzle requires using the digits 1 through 9 exactly once to satisfy three conditions involving logarithms. The solution is found by understanding the properties that make a logarithm an integer, rational, or irrational number. The most restrictive [&hellip;]<\/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-271992","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/271992","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=271992"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/271992\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=271992"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=271992"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=271992"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}