{"id":1143,"date":"2025-07-23T16:27:12","date_gmt":"2025-07-23T16:27:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1143"},"modified":"2026-03-18T17:55:07","modified_gmt":"2026-03-18T17:55:07","slug":"logarithmic-properties-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-properties-learn-it-5\/","title":{"raw":"Logarithmic Properties: Learn It 5","rendered":"Logarithmic Properties: Learn It 5"},"content":{"raw":"

Using the Change-of-Base Formula for Logarithms<\/h2>\r\nMost calculators can only evaluate common logarithm ([latex]\\mathrm{log}[\/latex]) and natural logarithm ([latex]\\mathrm{ln}[\/latex]). In order to evaluate logarithms with a base other than [latex]10[\/latex] 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.\r\n\r\n
\r\n

change-of-base formula<\/h3>\r\nThe change-of-base formula<\/strong> can be used to evaluate a logarithm with any base.\r\n\r\nFor any positive real numbers [latex]M[\/latex], [latex]b[\/latex], and [latex]n[\/latex], where [latex]n\\ne 1 [\/latex] and [latex]b\\ne 1[\/latex],\r\n

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

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

and<\/p>\r\n

[latex]{\\mathrm{log}}_{b}M=\\frac{\\mathrm{log}M}{\\mathrm{log}b}[\/latex]<\/p>\r\n\r\n<\/section>[reveal-answer q=\"994155\"]Where Does the Change-of-Base Formula Come From?[\/reveal-answer]\r\n[hidden-answer a=\"994155\"]\r\n\r\nTo derive the change-of-base formula, we use the one-to-one<\/strong> property and power rule for logarithms<\/strong>.\r\n\r\nGiven any positive real numbers [latex]M[\/latex], [latex]b[\/latex], and [latex]n[\/latex], where [latex]n\\ne 1 [\/latex] and [latex]b\\ne 1[\/latex], we show\r\n

[latex]{\\mathrm{log}}_{b}M\\text{=}\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}[\/latex]<\/p>\r\nLet [latex]y={\\mathrm{log}}_{b}M[\/latex]. Converting to exponential form, we obtain [latex]{b}^{y}=M[\/latex]. It follows that:\r\n

[latex]\\begin{array}{l}{\\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{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n

How To: Given a logarithm Of 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]<\/strong>\r\n
    \r\n \t
  1. Determine the new base [latex]n[\/latex], 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 [latex]e[\/latex].<\/li>\r\n \t
  2. Rewrite the log as a quotient using the change-of-base formula:\r\n
      \r\n \t
    • The numerator of the quotient will be a logarithm with base [latex]n[\/latex]\u00a0and argument [latex]M[\/latex].<\/li>\r\n \t
    • The denominator of the quotient will be a logarithm with base [latex]n[\/latex]\u00a0and argument [latex]b[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>
      Change [latex]{\\mathrm{log}}_{5}3[\/latex] to a quotient of natural logarithms.[reveal-answer q=\"303162\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"303162\"]Because we will be expressing [latex]{\\mathrm{log}}_{5}3[\/latex] as a quotient of natural logarithms, the new base\u00a0[latex]n\u00a0= e[\/latex].We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument [latex]3[\/latex]. The denominator of the quotient will be the natural log with argument [latex]5[\/latex].\r\n

      [latex]\\begin{array}{l}{\\mathrm{log}}_{b}M\\hfill & =\\frac{\\mathrm{ln}M}{\\mathrm{ln}b}\\hfill \\\\ {\\mathrm{log}}_{5}3\\hfill & =\\frac{\\mathrm{ln}3}{\\mathrm{ln}5}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      [ohm_question hide_question_numbers=1]321514[\/ohm_question]<\/section>
      Even if your calculator has a logarithm function for bases other than [latex]10[\/latex] or [latex]e[\/latex], you should become familiar with the change-of-base formula. Being able to manipulate formulas by hand is a useful skill in any quantitative or STEM-related field.<\/section>
      Evaluate [latex]{\\mathrm{log}}_{2}\\left(10\\right)[\/latex] using the change-of-base formula with a calculator.[reveal-answer q=\"448676\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"448676\"]According to the change-of-base formula, we can rewrite the log base [latex]2[\/latex] as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm which is the log base [latex]e[\/latex].\r\n
      [latex]\\begin{array}{l}{\\mathrm{log}}_{2}10=\\frac{\\mathrm{ln}10}{\\mathrm{ln}2}\\hfill & \\text{Apply the change of base formula using base }e.\\hfill \\\\ \\approx 3.3219\\hfill & \\text{Use a calculator to evaluate to 4 decimal places}.\\hfill \\end{array}[\/latex]<\/center>[\/hidden-answer]<\/section>
      [ohm_question hide_question_numbers=1]321517[\/ohm_question]<\/section>
      The first graphing calculators were programmed to only handle logarithms with base 10. One clever way to create the graph of a logarithm with a different base was to change the base of the logarithm using the principles from this section.Use an online graphing tool to plot [latex]f(x)=\\frac{\\log_{10}{x}}{\\log_{10}{2}}[\/latex].Follow these steps to see a clever way to graph a logarithmic function with base other than 10 on a graphing tool that only knows base 10.\r\n
        \r\n \t
      • Enter the function [latex]g(x) = \\log_{2}{x}[\/latex]<\/li>\r\n \t
      • Can you tell the difference between the graph of this function and the graph of [latex]f(x)[\/latex]? Explain what you think is happening.<\/li>\r\n \t
      • Your challenge is to write two new functions [latex]h(x),\\text{ and }k(x)[\/latex] that include a slider so you can change the base of the functions. Remember that there are restrictions on what values the base of a logarithm can take. You can click on the endpoints of the slider to change the input values.<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"

        Using the Change-of-Base Formula for Logarithms<\/h2>\n

        Most calculators can only evaluate common logarithm ([latex]\\mathrm{log}[\/latex]) and natural logarithm ([latex]\\mathrm{ln}[\/latex]). In order to evaluate logarithms with a base other than [latex]10[\/latex] 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>\n

        \n

        change-of-base formula<\/h3>\n

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

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

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

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

        and<\/p>\n

        [latex]{\\mathrm{log}}_{b}M=\\frac{\\mathrm{log}M}{\\mathrm{log}b}[\/latex]<\/p>\n<\/section>\n