{"id":1145,"date":"2025-07-23T16:26:50","date_gmt":"2025-07-23T16:26:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1145"},"modified":"2026-03-18T17:27:21","modified_gmt":"2026-03-18T17:27:21","slug":"logarithmic-properties-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-properties-learn-it-3\/","title":{"raw":"Logarithmic Properties: Learn It 3","rendered":"Logarithmic Properties: Learn It 3"},"content":{"raw":"

Using the Power Rule for Logarithms<\/h2>\r\nWe have explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as [latex]{x}^{2}[\/latex]? One method is as follows:\r\n

[latex]\\begin{array}{l}{\\mathrm{log}}_{b}\\left({x}^{2}\\right)\\hfill & ={\\mathrm{log}}_{b}\\left(x\\cdot x\\right)\\hfill \\\\ \\hfill & ={\\mathrm{log}}_{b}x+{\\mathrm{log}}_{b}x\\hfill \\\\ \\hfill & =2{\\mathrm{log}}_{b}x\\hfill \\end{array}[\/latex]<\/p>\r\nNotice that we used the product rule for logarithms<\/strong> to find a solution for the example above. By doing so, we have derived the power rule for logarithms<\/strong>, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that although the input to a logarithm may not be written as a power, we may be able to change it to a power.\r\n\r\n

\r\n

the power rule for logarithms<\/h3>\r\nThe power rule for logarithms<\/strong> can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.\r\n

[latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/p>\r\n\r\n<\/section>

How To: Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm<\/strong>\r\n
    \r\n \t
  1. Express the argument as a power, if needed.<\/li>\r\n \t
  2. Write the equivalent expression by multiplying the exponent times the logarithm of the base.<\/li>\r\n<\/ol>\r\n<\/section>
    \r\n
    The rules and properties of exponents will appear frequently when manipulating logarithms.<\/div>\r\n
    <\/div>\r\n
      \r\n \t
    • Product Rule<\/strong> [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/li>\r\n \t
    • Quotient Rule<\/strong> [latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/li>\r\n \t
    • Power Rule<\/strong> [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/li>\r\n \t
    • Zero Exponent<\/strong> [latex]{a}^{0}=1[\/latex]<\/li>\r\n \t
    • Negative Exponent<\/strong> [latex]{a}^{-n}=\\dfrac{1}{{a}^{n}} \\text{ and } {a}^{n}=\\dfrac{1}{{a}^{-n}}[\/latex]<\/li>\r\n \t
    • Power of a Product<\/strong> [latex]\\large{\\left(ab\\right)}^{n}={a}^{n}{b}^{n}[\/latex]<\/li>\r\n \t
    • Power of a Quotient<\/strong> [latex]\\large{\\left(\\dfrac{a}{b}\\right)}^{n}=\\dfrac{{a}^{n}}{{b}^{n}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>
      Rewrite [latex]{\\mathrm{log}}_{2}({x}^{5})[\/latex].[reveal-answer q=\"979765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"979765\"]The argument is already written as a power, so we identify the exponent, [latex]5[\/latex], and the base, x, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.\r\n

      [latex]{\\mathrm{log}}_{2}\\left({x}^{5}\\right)=5{\\mathrm{log}}_{2}x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      Rewrite [latex]{\\mathrm{log}}_{3}\\left(25\\right)[\/latex] using the power rule for logs.[reveal-answer q=\"984289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"984289\"]Expressing the argument as a power, we get [latex]{\\mathrm{log}}_{3}\\left(25\\right)={\\mathrm{log}}_{3}\\left({5}^{2}\\right)[\/latex].Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.\r\n

      [latex]{\\mathrm{log}}_{3}\\left({5}^{2}\\right)=2{\\mathrm{log}}_{3}\\left(5\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      \r\n
      \r\n\r\n[ohm_question hide_question_numbers=1]321500[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section>
      \r\n
      \r\n\r\n[ohm_question hide_question_numbers=1]321501[\/ohm_question]\r\n\r\n<\/div>\r\n<\/section>","rendered":"

      Using the Power Rule for Logarithms<\/h2>\n

      We have explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as [latex]{x}^{2}[\/latex]? One method is as follows:<\/p>\n

      [latex]\\begin{array}{l}{\\mathrm{log}}_{b}\\left({x}^{2}\\right)\\hfill & ={\\mathrm{log}}_{b}\\left(x\\cdot x\\right)\\hfill \\\\ \\hfill & ={\\mathrm{log}}_{b}x+{\\mathrm{log}}_{b}x\\hfill \\\\ \\hfill & =2{\\mathrm{log}}_{b}x\\hfill \\end{array}[\/latex]<\/p>\n

      Notice that we used the product rule for logarithms<\/strong> to find a solution for the example above. By doing so, we have derived the power rule for logarithms<\/strong>, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that although the input to a logarithm may not be written as a power, we may be able to change it to a power.<\/p>\n

      \n

      the power rule for logarithms<\/h3>\n

      The power rule for logarithms<\/strong> can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.<\/p>\n

      [latex]{\\mathrm{log}}_{b}\\left({M}^{n}\\right)=n{\\mathrm{log}}_{b}M[\/latex]<\/p>\n<\/section>\n

      How To: Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm<\/strong><\/p>\n
        \n
      1. Express the argument as a power, if needed.<\/li>\n
      2. Write the equivalent expression by multiplying the exponent times the logarithm of the base.<\/li>\n<\/ol>\n<\/section>\n
        \n
        The rules and properties of exponents will appear frequently when manipulating logarithms.<\/div>\n
        <\/div>\n
          \n
        • Product Rule<\/strong> [latex]{a}^{m}\\cdot {a}^{n}={a}^{m+n}[\/latex]<\/li>\n
        • Quotient Rule<\/strong> [latex]\\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[\/latex]<\/li>\n
        • Power Rule<\/strong> [latex]{\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}[\/latex]<\/li>\n
        • Zero Exponent<\/strong> [latex]{a}^{0}=1[\/latex]<\/li>\n
        • Negative Exponent<\/strong> [latex]{a}^{-n}=\\dfrac{1}{{a}^{n}} \\text{ and } {a}^{n}=\\dfrac{1}{{a}^{-n}}[\/latex]<\/li>\n
        • Power of a Product<\/strong> [latex]\\large{\\left(ab\\right)}^{n}={a}^{n}{b}^{n}[\/latex]<\/li>\n
        • Power of a Quotient<\/strong> [latex]\\large{\\left(\\dfrac{a}{b}\\right)}^{n}=\\dfrac{{a}^{n}}{{b}^{n}}[\/latex]<\/li>\n<\/ul>\n<\/section>\n
          Rewrite [latex]{\\mathrm{log}}_{2}({x}^{5})[\/latex].<\/p>\n