{"id":3535,"date":"2016-08-05T05:30:12","date_gmt":"2016-08-05T05:30:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&p=3535"},"modified":"2019-07-24T21:34:13","modified_gmt":"2019-07-24T21:34:13","slug":"convert-from-exponential-to-logarithmic-form","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/convert-from-exponential-to-logarithmic-form\/","title":{"raw":"Change of Base","rendered":"Change of Base"},"content":{"raw":"
\r\n
\r\n

Learning Outcome<\/h3>\r\n
    \r\n \t
  • Change the base of logarithmic expressions into base 10 or base e<\/em><\/li>\r\n<\/ul>\r\n<\/div>\r\n

    Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than\u00a0[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>\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 [latex]{\\mathrm{log}}_{b}M\\text{=}\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}[\/latex].<\/p>\r\n

    Let [latex]y={\\mathrm{log}}_{b}M[\/latex]. Rewriting in exponential form, we have\u00a0[latex]{b}^{y}=M[\/latex]. We can then take the log base [latex]n[\/latex] of both sides of the equation. It follows that:<\/p>\r\n\r\n

    [latex]\\begin{array}{c}{\\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]<\/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{array}{c}{\\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{array}[\/latex]<\/div>\r\n
    <\/div>\r\n
    Let us practice changing the base of a logarithmic expression from\u00a0[latex]5[\/latex] to base\u00a0<\/span>e.<\/em><\/div>\r\n<\/section>\r\n
    \r\n
    \r\n

    Example<\/h3>\r\nChange [latex]{\\mathrm{log}}_{5}3[\/latex] to a quotient of natural logarithms.\r\n[reveal-answer q=\"238811\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"238811\"]\r\n

    Because we will be expressing [latex]{\\mathrm{log}}_{5}3[\/latex] as a quotient of natural logarithms, the new base [latex]n=e[\/latex].<\/p>\r\n

    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\u00a0[latex]3[\/latex]. The denominator of the quotient will be the natural log with argument\u00a0[latex]5[\/latex].<\/p>\r\n\r\n

    [latex]\\begin{array}{c}{\\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]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can generalize the change of base formula in the following way:\r\n\r\n<\/div>\r\n
    \r\n
    \r\n

    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
    As we stated earlier, the main reason for changing the base of a logarithm is to be able to evaluate it with a calculator. In the following example, we will use the change of base formula on a logarithmic expression, and then evaluate the result with a calculator.<\/div>\r\n
    \r\n
    \r\n

    Example<\/h3>\r\nEvaluate [latex]{\\mathrm{log}}_{2}\\left(10\\right)[\/latex] using the change-of-base formula with a calculator.\r\n[reveal-answer q=\"385966\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"385966\"]\r\n

    According to the change-of-base formula, we can rewrite the log base\u00a0[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 e<\/i>.<\/p>\r\n

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

    \r\n

    Think About It<\/h3>\r\nCan we change common logarithms to natural logarithms?\r\n\r\nWrite your ideas in the textbox below before looking at the solution.\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"452135\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"452135\"]\r\n\r\nYes. Remember that [latex]\\mathrm{log}9[\/latex] means [latex]{\\text{log}}_{\\text{10}}\\text{9}[\/latex]. So, [latex]\\mathrm{log}9=\\frac{\\mathrm{ln}9}{\\mathrm{ln}10}[\/latex].<\/em>[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show more examples using the change-of-base formula to evaluate logarithms.\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=9kKg19s5b78[\/embed]\r\n

    Summary<\/h2>\r\nFor practical purposes found in many different sciences or finance applications, you may want to evaluate a logarithm with a calculator. The change of base formula will allow you to change the base of any logarithm to either\u00a0[latex]10[\/latex] or\u00a0e\u00a0<\/em>so you can evaluate it with a calculator. Here we have summarized the steps for using the change of base formula to evaluate a logarithm with the form [latex]{\\mathrm{log}}_{b}M[\/latex].\r\n
      \r\n \t
    1. Determine the new base n<\/em>, remembering that the common log, [latex]\\mathrm{log}\\left(x\\right)[\/latex], has base\u00a0[latex]10[\/latex], 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
        \r\n \t
      • The numerator of the quotient will be a logarithm with base n<\/em>\u00a0and argument M<\/em>.<\/li>\r\n \t
      • The denominator of the quotient will be a logarithm with base n<\/em>\u00a0and argument b<\/em>.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section>","rendered":"
        \n
        \n

        Learning Outcome<\/h3>\n
          \n
        • Change the base of logarithmic expressions into base 10 or base e<\/em><\/li>\n<\/ul>\n<\/div>\n

          Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than\u00a0[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

          To derive the change-of-base formula, we use the one-to-one<\/strong> property and power rule for logarithms<\/strong>.<\/p>\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 [latex]{\\mathrm{log}}_{b}M\\text{=}\\frac{{\\mathrm{log}}_{n}M}{{\\mathrm{log}}_{n}b}[\/latex].<\/p>\n

          Let [latex]y={\\mathrm{log}}_{b}M[\/latex]. Rewriting in exponential form, we have\u00a0[latex]{b}^{y}=M[\/latex]. We can then take the log base [latex]n[\/latex] of both sides of the equation. It follows that:<\/p>\n

          [latex]\\begin{array}{c}{\\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]<\/div>\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>\n

          [latex]\\begin{array}{c}{\\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{array}[\/latex]<\/div>\n
          <\/div>\n
          Let us practice changing the base of a logarithmic expression from\u00a0[latex]5[\/latex] to base\u00a0<\/span>e.<\/em><\/div>\n<\/section>\n
          \n
          \n

          Example<\/h3>\n

          Change [latex]{\\mathrm{log}}_{5}3[\/latex] to a quotient of natural logarithms.<\/p>\n

          Show Solution<\/span><\/p>\n
          \n

          Because we will be expressing [latex]{\\mathrm{log}}_{5}3[\/latex] as a quotient of natural logarithms, the new base [latex]n=e[\/latex].<\/p>\n

          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\u00a0[latex]3[\/latex]. The denominator of the quotient will be the natural log with argument\u00a0[latex]5[\/latex].<\/p>\n

          [latex]\\begin{array}{c}{\\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]<\/div>\n<\/div>\n<\/div>\n<\/div>\n

          We can generalize the change of base formula in the following way:<\/p>\n<\/div>\n

          \n
          \n

          The 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 M<\/em>, b<\/em>, and n<\/em>, 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].<\/div>\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]<\/div>\n

          and<\/p>\n

          [latex]{\\mathrm{log}}_{b}M=\\frac{\\mathrm{log}M}{\\mathrm{log}b}[\/latex]<\/div>\n<\/div>\n
          As we stated earlier, the main reason for changing the base of a logarithm is to be able to evaluate it with a calculator. In the following example, we will use the change of base formula on a logarithmic expression, and then evaluate the result with a calculator.<\/div>\n
          \n
          \n

          Example<\/h3>\n

          Evaluate [latex]{\\mathrm{log}}_{2}\\left(10\\right)[\/latex] using the change-of-base formula with a calculator.<\/p>\n

          Show Solution<\/span><\/p>\n
          \n

          According to the change-of-base formula, we can rewrite the log base\u00a0[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 e<\/i>.<\/p>\n

          [latex]\\begin{array}{c}{\\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]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

          \n

          Think About It<\/h3>\n

          Can we change common logarithms to natural logarithms?<\/p>\n

          Write your ideas in the textbox below before looking at the solution.<\/p>\n