{"id":11312,"date":"2015-07-14T19:40:23","date_gmt":"2015-07-14T19:40:23","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=11312"},"modified":"2015-09-09T19:04:58","modified_gmt":"2015-09-09T19:04:58","slug":"convert-from-exponential-to-logarithmic-form","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/convert-from-exponential-to-logarithmic-form\/","title":{"raw":"Convert from exponential to logarithmic form","rendered":"Convert from exponential to logarithmic form"},"content":{"raw":"

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b<\/em>, exponent x<\/em>, and output y<\/em>. Then we write [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\r\n\r\n

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Example 2: Converting from Exponential Form to Logarithmic Form<\/h3>\r\n

Write the following exponential equations in logarithmic form.<\/p>\r\n\r\n

    \r\n\t
  1. [latex]{2}^{3}=8[\/latex]<\/li>\r\n\t
  2. [latex]{5}^{2}=25[\/latex]<\/li>\r\n\t
  3. [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\n

    First, identify the values of b<\/em>, y<\/em>, and x<\/em>. Then, write the equation in the form [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\r\n\r\n

      \r\n\t
    1. [latex]{2}^{3}=8[\/latex]\r\n

      Here, b\u00a0<\/em>= 2, x\u00a0<\/em>= 3, and y\u00a0<\/em>= 8. Therefore, the equation [latex]{2}^{3}=8[\/latex] is equivalent to [latex]{\\mathrm{log}}_{2}\\left(8\\right)=3[\/latex].<\/p>\r\n<\/li>\r\n\t

    2. [latex]{5}^{2}=25[\/latex]\r\n

      Here, b\u00a0<\/em>= 5, x\u00a0<\/em>= 2, and y\u00a0<\/em>= 25. Therefore, the equation [latex]{5}^{2}=25[\/latex] is equivalent to [latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex].<\/p>\r\n<\/li>\r\n\t

    3. [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]\r\n

      Here, b\u00a0<\/em>= 10, x\u00a0<\/em>= \u20134, and [latex]y=\\frac{1}{10,000}[\/latex]. Therefore, the equation [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex] is equivalent to [latex]{\\text{log}}_{10}\\left(\\frac{1}{10,000}\\right)=-4[\/latex].<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

      \r\n

      Try It 2<\/h3>\r\n

      Write the following exponential equations in logarithmic form.<\/p>\r\n

      a. [latex]{3}^{2}=9[\/latex]<\/p>\r\n

      b. [latex]{5}^{3}=125[\/latex]<\/p>\r\n

      c. [latex]{2}^{-1}=\\frac{1}{2}[\/latex]<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

      To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b<\/em>, exponent x<\/em>, and output y<\/em>. Then we write [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\n

      \n
      \n
      \n

      Example 2: Converting from Exponential Form to Logarithmic Form<\/h3>\n

      Write the following exponential equations in logarithmic form.<\/p>\n

        \n
      1. [latex]{2}^{3}=8[\/latex]<\/li>\n
      2. [latex]{5}^{2}=25[\/latex]<\/li>\n
      3. [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]<\/li>\n<\/ol>\n<\/div>\n
        \n

        Solution<\/h3>\n

        First, identify the values of b<\/em>, y<\/em>, and x<\/em>. Then, write the equation in the form [latex]x={\\mathrm{log}}_{b}\\left(y\\right)[\/latex].<\/p>\n

          \n
        1. [latex]{2}^{3}=8[\/latex]\n

          Here, b\u00a0<\/em>= 2, x\u00a0<\/em>= 3, and y\u00a0<\/em>= 8. Therefore, the equation [latex]{2}^{3}=8[\/latex] is equivalent to [latex]{\\mathrm{log}}_{2}\\left(8\\right)=3[\/latex].<\/p>\n<\/li>\n

        2. [latex]{5}^{2}=25[\/latex]\n

          Here, b\u00a0<\/em>= 5, x\u00a0<\/em>= 2, and y\u00a0<\/em>= 25. Therefore, the equation [latex]{5}^{2}=25[\/latex] is equivalent to [latex]{\\mathrm{log}}_{5}\\left(25\\right)=2[\/latex].<\/p>\n<\/li>\n

        3. [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex]\n

          Here, b\u00a0<\/em>= 10, x\u00a0<\/em>= \u20134, and [latex]y=\\frac{1}{10,000}[\/latex]. Therefore, the equation [latex]{10}^{-4}=\\frac{1}{10,000}[\/latex] is equivalent to [latex]{\\text{log}}_{10}\\left(\\frac{1}{10,000}\\right)=-4[\/latex].<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n

          \n

          Try It 2<\/h3>\n

          Write the following exponential equations in logarithmic form.<\/p>\n

          a. [latex]{3}^{2}=9[\/latex]<\/p>\n

          b. [latex]{5}^{3}=125[\/latex]<\/p>\n

          c. [latex]{2}^{-1}=\\frac{1}{2}[\/latex]<\/p>\n

          Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

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