{"id":15978,"date":"2019-12-05T05:17:12","date_gmt":"2019-12-05T05:17:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/use-common-logarithms\/"},"modified":"2019-12-05T05:17:31","modified_gmt":"2019-12-05T05:17:31","slug":"use-common-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/use-common-logarithms\/","title":{"raw":"Evaluate Logarithms","rendered":"Evaluate Logarithms"},"content":{"raw":"\n
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Learning Outcome<\/h3>\n
    \n \t
  • Evaluate logarithms<\/li>\n<\/ul>\n<\/div>\n

    Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\\mathrm{log}}_{2}8[\/latex]. We ask, \"To what exponent must [latex]2[\/latex] be raised in order to get [latex]8[\/latex]?\" Because we already know [latex]{2}^{3}=8[\/latex], it follows that [latex]{\\mathrm{log}}_{2}8=3[\/latex].<\/p>\n

    Now consider solving [latex]{\\mathrm{log}}_{7}49[\/latex] and [latex]{\\mathrm{log}}_{3}27[\/latex] mentally.<\/p>\n\n

      \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>\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>\n<\/ul>\n

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

        \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>\n<\/ul>\nIn our first example, we will evaluate logarithms mentally.\n
        \n

        Example<\/h3>\nSolve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.\n[reveal-answer q=\"161686\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"161686\"]\n

        First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[\/latex]. Next, we ask, \"To what exponent must [latex]4[\/latex] be raised in order to get [latex]64[\/latex]?\"<\/p>\nWe know\n

        [latex]{4}^{3}=64[\/latex]<\/div>\n

        Therefore,<\/p>\n\n

        [latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]<\/div>\n[\/hidden-answer]\n\n<\/div>\nIn our first video, we will show more examples of evaluating logarithms mentally; this helps you get familiar with what a logarithm represents.\n\nhttps:\/\/youtu.be\/dxj5J9OpWGA\n\nIn our next example, we will evaluate the logarithm of a reciprocal.\n
        \n

        Example<\/h3>\nEvaluate [latex]y={\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)[\/latex] without using a calculator.\n[reveal-answer q=\"534439\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"534439\"]\n

        First we rewrite the logarithm in exponential form: [latex]{3}^{y}=\\frac{1}{27}[\/latex]. Next, we ask, \"To what exponent must [latex]3[\/latex] be raised in order to get [latex]\\frac{1}{27}[\/latex]?\"<\/p>\n

        We know [latex]{3}^{3}=27[\/latex], but what must we do to get the reciprocal, [latex]\\frac{1}{27}[\/latex]? Recall from working with exponents that [latex]{b}^{-a}=\\frac{1}{{b}^{a}}[\/latex]. We use this information to write:<\/p>\n

        [latex]\\begin{array}{ll}{3}^{-3} & =\\frac{1}{{3}^{3}} \\\\ & =\\frac{1}{27}\\hfill \\end{array}[\/latex]<\/p>\n

        Therefore, [latex]{\\mathrm{log}}_{3}\\left(\\frac{1}{27}\\right)=-3[\/latex].<\/p>\n[\/hidden-answer]\n\n<\/div>\n

        \n\n \n

        How To: Given a logarithm of the form [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], evaluate it mentally<\/h3>\n
          \n \t
        1. Rewrite the argument x<\/em> as a power of b<\/em>: [latex]{b}^{y}=x[\/latex].<\/li>\n \t
        2. Use previous knowledge of powers of b<\/em> to identify y<\/em> by asking, \"To what exponent should b<\/em> be raised in order to get x<\/em>?\"<\/li>\n<\/ol>\n

          <\/p>\n\n<\/div>\n

           Natural Logarithms<\/h2>\nThe most frequently used base for logarithms is <\/span>e<\/em>. Base <\/span>e<\/em> logarithms are important in calculus and some scientific applications; they are called <\/span>natural logarithms<\/strong>. The base <\/span>e<\/em> logarithm, [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex], has its own notation, [latex]\\mathrm{ln}\\left(x\\right)[\/latex].<\/span>\n

          Most values of [latex]\\mathrm{ln}\\left(x\\right)[\/latex] can be found only using a calculator. The major exception is that, because the logarithm of [latex]1[\/latex] is always [latex]0[\/latex] in any base, [latex]\\mathrm{ln}1=0[\/latex]. For other natural logarithms, we can use the [latex]\\mathrm{ln}[\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e<\/em> using the inverse property of logarithms.<\/p>\n\n

          \n

          A General Note: Definition of the Natural Logarithm<\/h3>\n

          A natural logarithm<\/strong> is a logarithm with base e<\/em>. We write [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] simply as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]. The natural logarithm of a positive number x<\/em> satisfies the following definition.<\/p>\n

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

          [latex]y=\\mathrm{ln}\\left(x\\right)\\text{ can be written as }{e}^{y}=x[\/latex]<\/div>\n

          We read [latex]\\mathrm{ln}\\left(x\\right)[\/latex] as, \"the logarithm with base e<\/em> of x<\/em>\" or \"the natural logarithm of x<\/em>.\"<\/p>\n

          The logarithm y<\/em> is the exponent to which e<\/em> must be raised to get x<\/em>.<\/p>\n

          Since the functions [latex]y=e{}^{x}[\/latex] and [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex] are inverse functions, [latex]\\mathrm{ln}\\left({e}^{x}\\right)=x[\/latex] for all x<\/em> and [latex]e{}^{\\mathrm{ln}\\left(x\\right)}=x[\/latex] for x <\/em>> [latex]0[\/latex].<\/p>\n\n<\/div>\nIn the next example, we will evaluate a natural logarithm using a calculator.\n

          \n

          Example<\/h3>\nEvaluate [latex]y=\\mathrm{ln}\\left(500\\right)[\/latex] to four decimal places using a calculator.\n[reveal-answer q=\"957920\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"957920\"]\n
            \n \t
          • Press [LN]<\/strong>.<\/li>\n \t
          • Enter [latex]500[\/latex], followed by [ ) ]<\/strong>.<\/li>\n \t
          • Press [ENTER]<\/strong>.<\/li>\n<\/ul>\n

            Rounding to four decimal places, [latex]\\mathrm{ln}\\left(500\\right)\\approx 6.2146[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\nIn our next video, we show more examples of how to evaluate natural logarithms using a calculator.\n\nhttps:\/\/youtu.be\/Rpounu3epSc\n

            Common Logarithms<\/h2>\nSometimes we may see a logarithm written without a base. In this case, we assume that the base is [latex]10[\/latex]. In other words, the expression [latex]{\\mathrm{log}}_{}[\/latex] means [latex]{\\mathrm{log}}_{10}[\/latex]. We call a base-[latex]10[\/latex] logarithm a common logarithm<\/strong>. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.\n
            \n

            Definition of Common Logarithm: Log is an exponent<\/h3>\nA common logarithm is a logarithm with base [latex]10[\/latex]. We write [latex]{\\mathrm{log}}_{10}(x)[\/latex]  simply as [latex]{\\mathrm{log}}_{}(x)[\/latex].  The common logarithm of a positive number, x, satisfies the following definition:\n\nFor [latex]x\\gt0[\/latex],\n

            [latex]y={\\mathrm{log}}_{}(x)[\/latex] can be written as [latex]10^y=x[\/latex]<\/p>\n

            We read [latex]{\\mathrm{log}}_{}(x)[\/latex] as \" the logarithm with base [latex]10[\/latex] of x\" or \"log base [latex]10[\/latex] of x\".<\/p>\n

            The logarithm y is the exponent to which 10 must be raised to get x.<\/p>\n\n<\/div>\n

            \n

            Example<\/h3>\nEvaluate [latex]{\\mathrm{log}}_{}(1000)[\/latex] without using a calculator.\n[reveal-answer q=\"80362\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"80362\"]We know [latex]10^3=1000[\/latex], therefore\n\n[latex]{\\mathrm{log}}_{}(1000)=3[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n
            \n

            Example<\/h3>\nEvaluate [latex]y={\\mathrm{log}}_{}(321)[\/latex] to four decimal places using a calculator.\n[reveal-answer q=\"782139\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"782139\"]\n
              \n \t
            • Press [LOG]<\/strong>.<\/li>\n \t
            • Enter [latex]321[\/latex],<\/em> followed by [ ) ]<\/strong>.<\/li>\n \t
            • Press [ENTER]<\/strong>.<\/li>\n<\/ul>\nRounding to four decimal places, [latex]{\\mathrm{log}}_{}(321)\\approx2.5065[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\nIn our last example, we will use a logarithm to find the difference in magnitude of two different earthquakes.\n\n<\/section>\n
              \n

              Example<\/h3>\nThe amount of energy released from one earthquake was [latex]500[\/latex] times greater than the amount of energy released from another. The equation [latex]10^x=500[\/latex] represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?\n[reveal-answer q=\"735383\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"735383\"]\n\nWe begin by rewriting the exponential equation in logarithmic form.\n

              [latex]10^x=500[\/latex]<\/p>\n

              [latex]{\\mathrm{log}}_{}(500)=x[\/latex]<\/p>\n

              Next we evaluate the logarithm using a calculator:<\/p>\n\n

                \n \t
              • Press [LOG]<\/strong>.<\/li>\n \t
              • Enter [latex]500[\/latex] <\/span>followed by [ ) ]<\/strong>.<\/li>\n \t
              • Press [ENTER]<\/strong>.<\/li>\n \t
              • To the nearest thousandth, [latex]{\\mathrm{log}}_{}(500)\\approx2.699[\/latex]<\/span><\/span><\/li>\n<\/ul>\n[\/hidden-answer]\n\n<\/div>\n

                Summary<\/h2>\nKnowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally because the logarithm is an exponent. Logarithms most commonly use base 10 and natural logarithms use base e. <\/em>Logarithms can also be evaluated with most kinds of calculator.\n","rendered":"
                \n
                \n

                Learning Outcome<\/h3>\n
                  \n
                • Evaluate logarithms<\/li>\n<\/ul>\n<\/div>\n

                  Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]{\\mathrm{log}}_{2}8[\/latex]. We ask, “To what exponent must [latex]2[\/latex] be raised in order to get [latex]8[\/latex]?” Because we already know [latex]{2}^{3}=8[\/latex], it follows that [latex]{\\mathrm{log}}_{2}8=3[\/latex].<\/p>\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 us 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

                      In our first example, we will evaluate logarithms mentally.<\/p>\n

                      \n

                      Example<\/h3>\n

                      Solve [latex]y={\\mathrm{log}}_{4}\\left(64\\right)[\/latex] without using a calculator.<\/p>\n

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

                      First we rewrite the logarithm in exponential form: [latex]{4}^{y}=64[\/latex]. Next, we ask, “To what exponent must [latex]4[\/latex] be raised in order to get [latex]64[\/latex]?”<\/p>\n

                      We know<\/p>\n

                      [latex]{4}^{3}=64[\/latex]<\/div>\n

                      Therefore,<\/p>\n

                      [latex]\\mathrm{log}{}_{4}\\left(64\\right)=3[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n

                      In our first video, we will show more examples of evaluating logarithms mentally; this helps you get familiar with what a logarithm represents.<\/p>\n