{"id":4037,"date":"2021-08-15T10:41:51","date_gmt":"2021-08-15T10:41:51","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4037"},"modified":"2021-11-26T16:46:11","modified_gmt":"2021-11-26T11:16:11","slug":"logs-change-of-base-formula","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/logs-change-of-base-formula\/","title":{"rendered":"Logs Change of Base Formula | Logarithm"},"content":{"rendered":"

Here you will learn logs change of base formula in logarithm and logarithmic inequalities with example.<\/p>\n

Let’s begin –<\/p>\n

Logs Change of Base Formula – Base Changing Theorem<\/h2>\n

It can be stated as “quotient of the algorithm of two numbers is independent of their common base.”<\/p>\n

\n

Symbolically, \\(log_b m\\) = \\(log_a m \\over log_a b\\), where a > 0 , \\(a\\neq1\\), b > 0, \\(b\\neq1\\).<\/p>\n<\/blockquote>\n\n\n

NOTE : <\/b><\/p>\n\t\t

\n\t\t (i)   \\(log_b a\\) . \\(log_a b\\) = \\(log a \\over log b\\) . \\(log b \\over log a\\) = 1 ;   Hence   \\(log_b a\\)=\\(1 \\over log_a b\\)<\/span>

\n\t\t (ii)   \\(a^{log_b c} \\) = \\(c^{log_b a} \\)

\n\t\t (iii)   \\(log_{a^k} m\\) = \\(1\\over k\\)\\(log_a m\\)          It is known as Base Power Formula.<\/b>

\n\t\t (iv)   The base of the logarithm can be any positive number other than 1, but in normal practice, only two bases are popular, these are 10 and e(=2.718 approx.). Logarithms of numbers to the base 10 are named as ‘common logarithm’ and the logarithms of numbers to the base e are called Natural or Napierian logarithm. We will consider logx as \\(log_e x\\) or lnx.<\/b>

<\/p>\n\n\n\n

Example : <\/span> Evaluate the given log : \\(81^{l\\over {log_5 3}}\\) + \\(27^{log_9 36}\\) + \\(3^{4\\over {log_7 9}}\\).<\/p>\n

Solution : <\/span>\\(81^{log_3 5}\\) + \\(3^{3log_9 36}\\) + \\(3^{4log_9 7}\\)
\n\\(\\implies\\) \\(3^{4log_3 5}\\) + \\(3^{log_3 {(36)}^{3\/2}}\\) + \\(3^{log_3 {7}^2}\\)
\n= 625 + 216 + 49
= 890.

<\/p>\n\n\n

Note :<\/strong><\/p>\n

\n

(i)  for a given value of N,\\(log_aN\\) will give us a unique value.<\/p>\n

(ii)  Logarithm of zero does not exist.<\/p>\n

(iii)  Logarithm of negative reals are not defined in the system of real numbers.<\/p>\n<\/blockquote>\n

Logarithmic Inequalities<\/h2>\n\n\n

(i)   \\(log_a x\\) < \\(log_a y\\)    <=>   [ x < y if a > 1 && x > y if 0 < a < 1 ]

\n (ii)   If a > 1, then     (a) \\(log_a x\\) < p => 0 < x < \\(a^p\\)             (b) \\(log_a x\\) > p => x > \\(a^p\\)

\n (iii)   If 0 < a < 1, then     (a) \\(log_a x\\) < p => x > \\(a^p\\)             (b) \\(log_a x\\) > p => 0 < x < \\(a^p\\)\n

<\/p>\n\n\n

Hope you learnt logs change of base formula in logarithm. To learn more practice more questions and get ahead in competition. Good Luck!<\/p>\n\n\n

\n
Next – Examples of logarithm<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – What are the properties of logarithm<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn logs change of base formula in logarithm and logarithmic inequalities with example. Let’s begin – Logs Change of Base Formula – Base Changing Theorem It can be stated as “quotient of the algorithm of two numbers is independent of their common base.” Symbolically, \\(log_b m\\) = \\(log_a m \\over log_a b\\), …<\/p>\n

Logs Change of Base Formula | Logarithm<\/span> Read More »<\/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":[13],"tags":[460,461,459],"yoast_head":"\nLogs Change of Base Formula | Logarithm - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn logs change of base formula in logarithm and logarithmic inequalities with example.\" \/>\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\/logs-change-of-base-formula\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Logs Change of Base Formula | Logarithm - 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