{"id":6836,"date":"2021-10-21T19:37:33","date_gmt":"2021-10-21T14:07:33","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6836"},"modified":"2021-10-25T00:48:52","modified_gmt":"2021-10-24T19:18:52","slug":"if-log_a-x-p-and-log_b-x2-q-then-log_x-sqrtab-is-equal-to","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/if-log_a-x-p-and-log_b-x2-q-then-log_x-sqrtab-is-equal-to\/","title":{"rendered":"If \\(log_a x\\) = p and \\(log_b {x^2}\\) = q then \\(log_x \\sqrt{ab}\\) is equal to"},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>\\(log_a x\\) = p \\(\\implies\\) \\(a^p\\) = x \\(\\implies\\) a = \\(x^{1\/p}\\)<\/p>\n<p>Similarly\u00a0 \\(b^q\\) = \\(x^2\\) \\(\\implies\\) b = \\(x^{2\/q}\\)<\/p>\n<p>Now, \\(log_x \\sqrt{ab}\\) = \\(log_x \\sqrt{x^{1\/p}x^{2\/q}}\\) = \\(log_x x^{({1\\over p}+{2\\over q}){1\\over 2}}\\) = \\(1\\over {2p}\\) + \\(1\\over q\\).<\/p>\n<hr \/>\n<h3 style=\"text-align: center;\">Similar Questions<\/h3>\n<p><a class=\"row-title\" style=\"color: #2271b1; text-decoration: none; cursor: pointer; transition: none 0s ease 0s; outline: 0px; overflow-wrap: break-word; font-weight: 600; font-size: 14px; font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: #ffffff;\" href=\"https:\/\/mathemerize.com\/solve-for-x-2x-2-1over-41over-x\/\" aria-label=\"\u201cSolve for x : \\(2^{x + 2}\\) &gt; \\(({1\\over 4})^{1\\over x}\\).\u201d (Edit)\">Solve for x : \\(2^{x + 2}\\) &gt; \\(({1\\over 4})^{1\\over x}\\).<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/evaluate-the-given-log-81lover-log_5-3-27log_9-36-34over-log_7-9\/\" aria-label=\"\u201cEvaluate the given log : \\(81^{l\\over {log_5 3}}\\) + \\(27^{log_9 36}\\) + \\(3^{4\\over {log_7 9}}\\).\u201d (Edit)\">Evaluate the given log : \\(81^{l\\over {log_5 3}}\\) + \\(27^{log_9 36}\\) + \\(3^{4\\over {log_7 9}}\\).<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/find-the-value-of-2log2over-5-3log25over-8-log625over-128\/\" aria-label=\"\u201cFind the value of \\(2log{2\\over 5}\\) + \\(3log{25\\over 8}\\) \u2013 \\(log{625\\over 128}\\).\u201d (Edit)\">Find the value of \\(2log{2\\over 5}\\) + \\(3log{25\\over 8}\\) \u2013 \\(log{625\\over 128}\\).<\/a><\/p>\n<p><a class=\"row-title\" href=\"https:\/\/mathemerize.com\/if-log_e-x-log_e-y-a-log_e-y-log_e-z-b-log_e-z-log_e-x-c-then-find-the-value-of-xover-yb-c-times-yover-zc-a\/\" aria-label=\"\u201cIf \\(log_e x\\) \u2013 \\(log_e y\\) = a, \\(log_e y\\) \u2013 \\(log_e z\\) = b &amp; \\(log_e z\\) \u2013 \\(log_e x\\) = c, then find the value of \\(({x\\over y})^{b-c}\\) \\(\\times\\) \\(({y\\over z})^{c-a}\\) \\(\\times\\) \\(({z\\over x})^{a-b}\\).\u201d (Edit)\">If \\(log_e x\\) \u2013 \\(log_e y\\) = a, \\(log_e y\\) \u2013 \\(log_e z\\) = b &amp; \\(log_e z\\) \u2013 \\(log_e x\\) = c, then find the value of \\(({x\\over y})^{b-c}\\) \\(\\times\\) \\(({y\\over z})^{c-a}\\) \\(\\times\\) \\(({z\\over x})^{a-b}\\).<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : \\(log_a x\\) = p \\(\\implies\\) \\(a^p\\) = x \\(\\implies\\) a = \\(x^{1\/p}\\) Similarly\u00a0 \\(b^q\\) = \\(x^2\\) \\(\\implies\\) b = \\(x^{2\/q}\\) Now, \\(log_x \\sqrt{ab}\\) = \\(log_x \\sqrt{x^{1\/p}x^{2\/q}}\\) = \\(log_x x^{({1\\over p}+{2\\over q}){1\\over 2}}\\) = \\(1\\over {2p}\\) + \\(1\\over q\\). Similar Questions Solve for x : \\(2^{x + 2}\\) &gt; \\(({1\\over 4})^{1\\over x}\\). Evaluate the &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/if-log_a-x-p-and-log_b-x2-q-then-log_x-sqrtab-is-equal-to\/\"> <span class=\"screen-reader-text\">If \\(log_a x\\) = p and \\(log_b {x^2}\\) = q then \\(log_x \\sqrt{ab}\\) is equal to<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[55,43],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>If \\(log_a x\\) = p and \\(log_b {x^2}\\) = q then \\(log_x \\sqrt{ab}\\) is equal to<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/if-log_a-x-p-and-log_b-x2-q-then-log_x-sqrtab-is-equal-to\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If \\(log_a x\\) = p and \\(log_b {x^2}\\) = q then \\(log_x \\sqrt{ab}\\) is equal to\" \/>\n<meta property=\"og:description\" content=\"Solution : (log_a x) = p (implies) (a^p) = x (implies) a = (x^{1\/p}) Similarly\u00a0 (b^q) = (x^2) (implies) b = (x^{2\/q}) Now, (log_x sqrt{ab}) = (log_x sqrt{x^{1\/p}x^{2\/q}}) = (log_x x^{({1over p}+{2over q}){1over 2}}) = (1over {2p}) + (1over q). 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