{"id":1064,"date":"2025-07-22T00:06:44","date_gmt":"2025-07-22T00:06:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1064"},"modified":"2026-03-17T18:12:16","modified_gmt":"2026-03-17T18:12:16","slug":"logarithmic-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/logarithmic-functions-learn-it-2\/","title":{"raw":"Logarithmic Functions: Learn It 2","rendered":"Logarithmic Functions: Learn It 2"},"content":{"raw":"

Evaluating Logarithms<\/h2>\r\nKnowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. If we remember the logarithm is the exponent, it makes the conversion easier. You may want to repeat, \u201cbase to the exponent gives us the number.\u201d\r\n\r\n
Consider [latex]{\\mathrm{log}}_{2}8[\/latex].\r\n[latex]\\\\[\/latex]\r\nWe ask, \"To what exponent must [latex]2[\/latex] be raised in order to get [latex]8[\/latex]?\"\r\n\r\nBecause we already know [latex]{2}^{3}=8[\/latex], it follows that\r\n\r\n[latex]{\\mathrm{log}}_{2}8=3[\/latex].<\/section>
Now consider solving [latex]{\\mathrm{log}}_{7}49[\/latex] and [latex]{\\mathrm{log}}_{3}27[\/latex] mentally.\r\n
    \r\n \t
  • We ask, \"To what exponent must [latex]7[\/latex] be raised in order to get [latex]49[\/latex]?\" We know [latex]{7}^{2}=49[\/latex]. Therefore, [latex]{\\mathrm{log}}_{7}49=2[\/latex].<\/li>\r\n \t
  • We ask, \"To what exponent must [latex]3[\/latex] be raised in order to get [latex]27[\/latex]?\" We know [latex]{3}^{3}=27[\/latex]. Therefore, [latex]{\\mathrm{log}}_{3}27=3[\/latex].<\/li>\r\n<\/ul>\r\nEven some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.\r\n
      \r\n \t
    • We ask, \"To what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}[\/latex]? \" We know [latex]{2}^{2}=4[\/latex] and [latex]{3}^{2}=9[\/latex], so [latex]{\\left(\\frac{2}{3}\\right)}^{2}=\\frac{4}{9}[\/latex]. Therefore, [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2[\/latex].<\/li>\r\n<\/ul>\r\n
      It may be tempting to use your calculator to evaluate these logarithms but try to evaluate them mentally as it will aid your understanding of what a logarithm is, and will help you navigate more complicated situations.<\/section>
      How To: Given a logarithm of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], evaluate it mentally<\/strong>\r\n
        \r\n \t
      1. Rewrite the argument [latex]x[\/latex]\u00a0as a power of [latex]b[\/latex]: [latex]{b}^{y}=x[\/latex].<\/li>\r\n \t
      2. Use previous knowledge of powers of [latex]b[\/latex]\u00a0to identify [latex]y[\/latex]\u00a0by asking, \"To what exponent should [latex]b[\/latex]\u00a0be raised in order to get [latex]x[\/latex]?\"<\/li>\r\n<\/ol>\r\n<\/section><\/section>
        Solve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.[reveal-answer q=\"879580\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"879580\"]First we rewrite the logarithm in exponential form:\r\n\r\n[latex]{4}^{y}=64[\/latex].\r\n\r\nNext, we ask, \"To what exponent must [latex]4[\/latex] be raised in order to get [latex]64[\/latex]?\"\r\n\r\nWe know [latex]{4}^{3}=64[\/latex]\r\n\r\nTherefore,\r\n

        [latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

        Evaluate [latex]y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex] without using a calculator.[reveal-answer q=\"861965\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"861965\"]First we rewrite the logarithm in exponential form:\r\n\r\n[latex]{3}^{y}=\\frac{1}{27}[\/latex].\r\n\r\nNext, we ask, \"To what exponent must 3 be raised in order to get [latex]\\frac{1}{27}[\/latex]\"?\r\n[latex]\\\\[\/latex]\r\nWe know [latex]{3}^{3}=27[\/latex], but what must we do to get the reciprocal, [latex]\\frac{1}{27}[\/latex]?\r\n\r\nRecall from working with exponents that [latex]{b}^{-a}=\\frac{1}{{b}^{a}}[\/latex]. We use this information to write\r\n

        [latex]\\begin{array}{l}{3}^{-3}=\\frac{1}{{3}^{3}}=\\frac{1}{27}\\hfill \\end{array}[\/latex]<\/p>\r\nTherefore, [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

        [ohm_question hide_question_numbers=1]321428[\/ohm_question]<\/section>
        [ohm_question hide_question_numbers=1]321429[\/ohm_question]<\/section>","rendered":"

        Evaluating Logarithms<\/h2>\n

        Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. If we remember the logarithm is the exponent, it makes the conversion easier. You may want to repeat, \u201cbase to the exponent gives us the number.\u201d<\/p>\n

        Consider [latex]{\\mathrm{log}}_{2}8[\/latex].
        \n[latex]\\\\[\/latex]
        \nWe ask, “To what exponent must [latex]2[\/latex] be raised in order to get [latex]8[\/latex]?”<\/p>\n

        Because we already know [latex]{2}^{3}=8[\/latex], it follows that<\/p>\n

        [latex]{\\mathrm{log}}_{2}8=3[\/latex].<\/section>\n

        Now consider solving [latex]{\\mathrm{log}}_{7}49[\/latex] and [latex]{\\mathrm{log}}_{3}27[\/latex] mentally.<\/p>\n
          \n
        • We ask, “To what exponent must [latex]7[\/latex] be raised in order to get [latex]49[\/latex]?” We know [latex]{7}^{2}=49[\/latex]. Therefore, [latex]{\\mathrm{log}}_{7}49=2[\/latex].<\/li>\n
        • We ask, “To what exponent must [latex]3[\/latex] be raised in order to get [latex]27[\/latex]?” We know [latex]{3}^{3}=27[\/latex]. Therefore, [latex]{\\mathrm{log}}_{3}27=3[\/latex].<\/li>\n<\/ul>\n

          Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.<\/p>\n

            \n
          • We ask, “To what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}[\/latex]? ” We know [latex]{2}^{2}=4[\/latex] and [latex]{3}^{2}=9[\/latex], so [latex]{\\left(\\frac{2}{3}\\right)}^{2}=\\frac{4}{9}[\/latex]. Therefore, [latex]{\\mathrm{log}}_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2[\/latex].<\/li>\n<\/ul>\n
            It may be tempting to use your calculator to evaluate these logarithms but try to evaluate them mentally as it will aid your understanding of what a logarithm is, and will help you navigate more complicated situations.<\/section>\n
            How To: Given a logarithm of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], evaluate it mentally<\/strong><\/p>\n
              \n
            1. Rewrite the argument [latex]x[\/latex]\u00a0as a power of [latex]b[\/latex]: [latex]{b}^{y}=x[\/latex].<\/li>\n
            2. Use previous knowledge of powers of [latex]b[\/latex]\u00a0to identify [latex]y[\/latex]\u00a0by asking, “To what exponent should [latex]b[\/latex]\u00a0be raised in order to get [latex]x[\/latex]?”<\/li>\n<\/ol>\n<\/section>\n<\/section>\n
              Solve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.<\/p>\n