{"id":1063,"date":"2025-07-22T00:07:12","date_gmt":"2025-07-22T00:07:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1063"},"modified":"2026-03-17T18:14:23","modified_gmt":"2026-03-17T18:14:23","slug":"logarithmic-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-functions-learn-it-3\/","title":{"raw":"Logarithmic Functions: Learn It 3","rendered":"Logarithmic Functions: Learn It 3"},"content":{"raw":"
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Common Logarithms<\/h2>\r\nSometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expressions [latex]\\text{log}(x)[\/latex] means [latex]\\text{log}_{10}(x).[\/latex] We call a base-[latex]10[\/latex] logarithm a\u00a0common logarithm.\u00a0<\/strong>Common logarithms are used to measure the Richter Scale of earthquakes. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.\r\n\r\n
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common logarithm<\/h3>\r\nA common logarithm<\/strong> is a logarithm with base [latex]10[\/latex]. We write [latex]\\text{log}_{10}(x)[\/latex] simply as [latex]\\text{log}(x)[\/latex]. The common logarithm of a positive number [latex]x[\/latex] satisfies the following definition:\r\n\r\n \r\n\r\nFor [latex]x>0[\/latex],\r\n

[latex]y=\\text{log}(x)\\text{ is equivalent to }{10}^{y}=x[\/latex]<\/p>\r\n \r\n

We read [latex]\\text{log}(x)[\/latex] as, \"the logarithm with base [latex]10[\/latex]\u00a0of [latex]x[\/latex]\" or \"the common logarithm of [latex]x[\/latex].\"<\/p>\r\n\r\n

    \r\n \t
  • The logarithm [latex]y[\/latex] is the exponent to which [latex]10[\/latex] must be raised to get [latex]x[\/latex].<\/li>\r\n \t
  • Since the functions [latex]y=10^{x}[\/latex] and [latex]y=\\mathrm{log}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{log}\\left({10}^{x}\\right)=x[\/latex] for all [latex]x[\/latex] and [latex]10^{\\mathrm{log}\\left(x\\right)}=x[\/latex] for [latex]x>0[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>
    Evaluate [latex]\\mathrm{log}(100)[\/latex].\r\n\r\n
    \r\n\r\nTo evaluate [latex]\\mathrm{log}(100)[\/latex], we're looking for the power to which [latex]10[\/latex] must be raised to get [latex]100[\/latex]. In other words:\r\n\r\n
    [latex]\\begin{align*} \\log(100) &= x \\\\ 10^x &= 100 \\\\ 100 &= 10^2 \\\\ 10^x &= 10^2 \\\\ x &= 2 \\\\ \\log(100) &= 2 \\end{align*}[\/latex]<\/center><\/section>
    Evaluate [latex]\\mathrm{log}(110)[\/latex].To evaluate [latex]\\mathrm{log}(110)[\/latex], we need to use a calculator or logarithm tables since [latex]110[\/latex] is not a power of [latex]10[\/latex].[reveal-answer q=\"955300\"]Calculator Steps[\/reveal-answer]\r\n[hidden-answer a=\"955300\"]Given a common logarithm of the form [latex]y=\\mathrm{log}\\left(x\\right)[\/latex], evaluate it using a calculator\r\n
      \r\n \t
    1. Press [LOG]<\/strong>.<\/li>\r\n \t
    2. Enter the value given for [latex]x[\/latex], followed by [ ) ]<\/strong>.<\/li>\r\n \t
    3. Press [ENTER]<\/strong>.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"918162\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"918162\"][latex]\\mathrm{log}(110) \\approx 2.0414[\/latex] [\/hidden-answer]\r\n\r\n<\/section>
      [ohm_question hide_question_numbers=1]321430[\/ohm_question]<\/section>
      [ohm_question hide_question_numbers=1]321432[\/ohm_question]<\/section><\/section>","rendered":"
      \n

      Common Logarithms<\/h2>\n

      Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expressions [latex]\\text{log}(x)[\/latex] means [latex]\\text{log}_{10}(x).[\/latex] We call a base-[latex]10[\/latex] logarithm a\u00a0common logarithm.\u00a0<\/strong>Common logarithms are used to measure the Richter Scale of earthquakes. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.<\/p>\n

      \n

      common logarithm<\/h3>\n

      A common logarithm<\/strong> is a logarithm with base [latex]10[\/latex]. We write [latex]\\text{log}_{10}(x)[\/latex] simply as [latex]\\text{log}(x)[\/latex]. The common logarithm of a positive number [latex]x[\/latex] satisfies the following definition:<\/p>\n

       <\/p>\n

      For [latex]x>0[\/latex],<\/p>\n

      [latex]y=\\text{log}(x)\\text{ is equivalent to }{10}^{y}=x[\/latex]<\/p>\n

       <\/p>\n

      We read [latex]\\text{log}(x)[\/latex] as, “the logarithm with base [latex]10[\/latex]\u00a0of [latex]x[\/latex]” or “the common logarithm of [latex]x[\/latex].”<\/p>\n

        \n
      • The logarithm [latex]y[\/latex] is the exponent to which [latex]10[\/latex] must be raised to get [latex]x[\/latex].<\/li>\n
      • Since the functions [latex]y=10^{x}[\/latex] and [latex]y=\\mathrm{log}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{log}\\left({10}^{x}\\right)=x[\/latex] for all [latex]x[\/latex] and [latex]10^{\\mathrm{log}\\left(x\\right)}=x[\/latex] for [latex]x>0[\/latex].<\/li>\n<\/ul>\n<\/section>\n
        Evaluate [latex]\\mathrm{log}(100)[\/latex].<\/p>\n
        \n

        To evaluate [latex]\\mathrm{log}(100)[\/latex], we’re looking for the power to which [latex]10[\/latex] must be raised to get [latex]100[\/latex]. In other words:<\/p>\n

        [latex]\\begin{align*} \\log(100) &= x \\\\ 10^x &= 100 \\\\ 100 &= 10^2 \\\\ 10^x &= 10^2 \\\\ x &= 2 \\\\ \\log(100) &= 2 \\end{align*}[\/latex]<\/div>\n<\/section>\n
        Evaluate [latex]\\mathrm{log}(110)[\/latex].To evaluate [latex]\\mathrm{log}(110)[\/latex], we need to use a calculator or logarithm tables since [latex]110[\/latex] is not a power of [latex]10[\/latex].<\/p>\n