{"id":11212,"date":"2015-07-14T18:52:16","date_gmt":"2015-07-14T18:52:16","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11212"},"modified":"2015-09-09T21:13:22","modified_gmt":"2015-09-09T21:13:22","slug":"use-the-change-of-base-formula-for-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/use-the-change-of-base-formula-for-logarithms\/","title":{"raw":"Use the change-of-base formula for logarithms","rendered":"Use the change-of-base formula for logarithms"},"content":{"raw":"
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Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or [latex]e[\/latex], we use the change-of-base formula<\/strong> to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.<\/p>\r\n

To derive the change-of-base formula, we use the one-to-one<\/strong> property and power rule for logarithms<\/strong>.<\/p>\r\n

Given any positive real numbers M<\/em>, b<\/em>, and n<\/em>, where [latex]n\\ne 1 [\/latex] and [latex]b\\ne 1[\/latex], we show<\/p>\r\n\r\n

[latex]{\\mathrm{log}}_{b}M\\text{=}\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}[\/latex]<\/div>\r\n

Let [latex]y={\\mathrm{log}}_{b}M[\/latex]. By taking the log base [latex]n[\/latex] of both sides of the equation, we arrive at an exponential form, namely [latex]{b}^{y}=M[\/latex]. It follows that<\/p>\r\n\r\n

[latex]\\begin{cases}{\\mathrm{log}}_{n}\\left({b}^{y}\\right)\\hfill & ={\\mathrm{log}}_{n}M\\hfill & \\text{Apply the one-to-one property}.\\hfill \\\\ y{\\mathrm{log}}_{n}b\\hfill & ={\\mathrm{log}}_{n}M \\hfill & \\text{Apply the power rule for logarithms}.\\hfill \\\\ y\\hfill & =\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}\\hfill & \\text{Isolate }y.\\hfill \\\\ {\\mathrm{log}}_{b}M\\hfill & =\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}\\hfill & \\text{Substitute for }y.\\hfill \\end{cases}[\/latex]<\/div>\r\n

For example, to evaluate [latex]{\\mathrm{log}}_{5}36[\/latex] using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.<\/p>\r\n\r\n

[latex]\\begin{cases}{\\mathrm{log}}_{5}36\\hfill & =\\frac{\\mathrm{log}\\left(36\\right)}{\\mathrm{log}\\left(5\\right)}\\hfill & \\text{Apply the change of base formula using base 10}\\text{.}\\hfill \\\\ \\hfill & \\approx 2.2266\\text{ }\\hfill & \\text{Use a calculator to evaluate to 4 decimal places}\\text{.}\\hfill \\end{cases}[\/latex]<\/div>\r\n
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A General Note: The Change-of-Base Formula<\/h3>\r\n

The change-of-base formula<\/strong> can be used to evaluate a logarithm with any base.<\/p>\r\n

For any positive real numbers M<\/em>, b<\/em>, and n<\/em>, where [latex]n\\ne 1 [\/latex] and [latex]b\\ne 1[\/latex],<\/p>\r\n\r\n

[latex]{\\mathrm{log}}_{b}M\\text{=}\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}[\/latex].<\/div>\r\n

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.<\/p>\r\n\r\n

[latex]{\\mathrm{log}}_{b}M=\\frac{\\mathrm{ln}M}{\\mathrm{ln}b}[\/latex]<\/div>\r\n

and<\/p>\r\n\r\n

[latex]{\\mathrm{log}}_{b}M=\\frac{\\mathrm{log}M}{\\mathrm{log}b}[\/latex]<\/div>\r\n<\/div>\r\n
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How To: Given a logarithm with the form [latex]{\\mathrm{log}}_{b}M[\/latex], use the change-of-base formula to rewrite it as a quotient of logs with any positive base [latex]n[\/latex], where [latex]n\\ne 1[\/latex].<\/h3>\r\n
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  1. Determine the new base n<\/em>, remembering that the common log, [latex]\\mathrm{log}\\left(x\\right)[\/latex], has base 10, and the natural log, [latex]\\mathrm{ln}\\left(x\\right)[\/latex], has base e<\/em>.<\/li>\r\n\t
  2. Rewrite the log as a quotient using the change-of-base formula\r\n