{"id":1061,"date":"2025-07-22T00:08:05","date_gmt":"2025-07-22T00:08:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1061"},"modified":"2026-03-17T20:54:07","modified_gmt":"2026-03-17T20:54:07","slug":"graphs-of-logarithms-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-logarithms-learn-it-2\/","title":{"raw":"Graphs of Logarithms: Learn It 2","rendered":"Graphs of Logarithms: Learn It 2"},"content":{"raw":"

Graphing a Logarithmic Function Using a Table of Values<\/h2>\r\nNow that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] along with all of its transformations: shifts, stretches, compressions, and reflections.\r\n\r\n
Recall that if an invertible function [latex]f(x)[\/latex] contains a point, [latex]\\left(a, b\\right)[\/latex], then the inverse function [latex]f^{-1}\\left(x\\right)[\/latex] must contain the point [latex]\\left(b, a\\right)[\/latex].<\/section>
We begin with the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Because every logarithmic function of this form is the inverse of an exponential function of the form [latex]y={b}^{x}[\/latex], their graphs will be reflections of each other across the line [latex]y=x[\/latex].<\/section>
To illustrate this, we can observe the relationship between the input and output values of [latex]y={2}^{x}[\/latex] and its equivalent logarithmic form [latex]x={\\mathrm{log}}_{2}\\left(y\\right)[\/latex] in the table below.<\/section>
\r\n\r\n\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n[latex]\u20133[\/latex]<\/td>\r\n[latex]\u20132[\/latex]<\/td>\r\n[latex]\u20131[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]{2}^{x}=y[\/latex]<\/strong><\/td>\r\n[latex]\\frac{1}{8}[\/latex]<\/td>\r\n[latex]\\frac{1}{4}[\/latex]<\/td>\r\n[latex]\\frac{1}{2}[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\r\n[latex]\u20133[\/latex]<\/td>\r\n[latex]\u20132[\/latex]<\/td>\r\n[latex]\u20131[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing the inputs and outputs from the table above, we can build another table to observe the relationship between points on the graphs of the inverse functions [latex]f\\left(x\\right)={2}^{x}[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].\r\n\r\n\r\n\r\n\r\n
[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\r\n[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\r\n[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\r\n[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\r\n[latex]\\left(0,1\\right)[\/latex]<\/td>\r\n[latex]\\left(1,2\\right)[\/latex]<\/td>\r\n[latex]\\left(2,4\\right)[\/latex]<\/td>\r\n[latex]\\left(3,8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\r\n[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\r\n[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\r\n[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\r\n[latex]\\left(1,0\\right)[\/latex]<\/td>\r\n[latex]\\left(2,1\\right)[\/latex]<\/td>\r\n[latex]\\left(4,2\\right)[\/latex]<\/td>\r\n[latex]\\left(8,3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs we would expect, the [latex]x[\/latex]and [latex]y[\/latex]-coordinates are reversed for the inverse functions. The figure below\u00a0shows the graphs of [latex]f[\/latex]\u00a0and [latex]g[\/latex].\r\n\r\n[caption id=\"attachment_5016\" align=\"aligncenter\" width=\"408\"]\"Graph Notice that the graphs of [latex]f\\left(x\\right)={2}^{x}[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] are reflections about the line y = x since they are inverses of each other.[\/caption]Observe the following from the graph:\r\n
    \r\n \t
  • [latex]f\\left(x\\right)={2}^{x}[\/latex] has a y<\/em>-intercept at [latex]\\left(0,1\\right)[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] has an x<\/em>-intercept at [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t
  • The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex], [latex]\\left(-\\infty ,\\infty \\right)[\/latex], is the same as the range of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/li>\r\n \t
  • The range of [latex]f\\left(x\\right)={2}^{x}[\/latex], [latex]\\left(0,\\infty \\right)[\/latex], is the same as the domain of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/section><\/section>
    \r\n

    characteristics of the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nFor any real number [latex]x[\/latex] and constant [latex]b \\gt 0[\/latex], [latex]b\\ne 1[\/latex], we can see the following characteristics in the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]:\r\n
      \r\n \t
    • one-to-one function<\/li>\r\n \t
    • vertical asymptote: [latex]x\u00a0= 0[\/latex]<\/li>\r\n \t
    • domain: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\r\n \t
    • range: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/li>\r\n \t
    • [latex]x[\/latex]-<\/em>intercept: [latex]\\left(1,0\\right)[\/latex] and key point [latex]\\left(b,1\\right)[\/latex]<\/li>\r\n \t
    • [latex]y[\/latex]-intercept: none<\/li>\r\n \t
    • increasing if [latex]b \\gt 1[\/latex]<\/li>\r\n \t
    • decreasing if [latex]0 \\lt b \\lt 1[\/latex]<\/li>\r\n<\/ul>\r\n \r\n\r\n\"\"\r\n\r\n<\/section>
      How To: Given a logarithmic function Of the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph the function<\/strong>\r\n
        \r\n \t
      1. Draw and label the vertical asymptote, [latex]x = 0[\/latex].<\/li>\r\n \t
      2. Plot the [latex]x[\/latex]-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t
      3. Plot the key point [latex]\\left(b,1\\right)[\/latex].<\/li>\r\n \t
      4. Draw a smooth curve through the points.<\/li>\r\n \t
      5. State the domain, [latex]\\left(0,\\infty \\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote, [latex]x = 0[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
        Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.[reveal-answer q=\"909934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"909934\"]Before graphing, identify the behavior and key points for the graph.\r\n
          \r\n \t
        • [caption id=\"\" align=\"alignright\" width=\"324\"]\"Graph The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x = 0.[\/caption]Since [latex]b\u00a0= 5[\/latex] is greater than one, we know the function is increasing<\/strong>. The left tail of the graph will approach the vertical asymptote<\/strong> [latex]x\u00a0= 0[\/latex], and the right tail will increase slowly without bound.<\/li>\r\n \t
        • The [latex]x[\/latex]-intercept<\/strong> is [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t
        • The key point<\/strong> [latex]\\left(5,1\\right)[\/latex] is on the graph.<\/li>\r\n \t
        • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>
          Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{5}}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.[reveal-answer q=\"150661\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"150661\"]\r\n\"GraphSince [latex]b = \\frac{1}{5}[\/latex] is less than one, we know the function is decreasing<\/strong>. The left tail of the graph will approach the vertical asymptote<\/strong> [latex]x = 0[\/latex], and the right tail will decrease slowly without bound.\r\n
            \r\n \t
          • The [latex]x[\/latex]-intercept<\/strong> is [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t
          • The key point<\/strong> [latex]\\left(\\frac{1}{5},1\\right)[\/latex] is on the graph.<\/li>\r\n \t
          • Another key point<\/strong> on a decreasing graph is [latex](\\dfrac{1}{b},-1)[\/latex]. That is: [latex](\\frac{1}{\\frac{1}{5}}, -1) = (5,-1)[\/latex]<\/li>\r\n \t
          • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\r\n<\/ul>\r\nThe domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is [latex]x\u00a0= 0[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
            [ohm_question hide_question_numbers=1]321444[\/ohm_question]<\/section>
            [ohm_question hide_question_numbers=1]321445[\/ohm_question]<\/section>","rendered":"

            Graphing a Logarithmic Function Using a Table of Values<\/h2>\n

            Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] along with all of its transformations: shifts, stretches, compressions, and reflections.<\/p>\n

            Recall that if an invertible function [latex]f(x)[\/latex] contains a point, [latex]\\left(a, b\\right)[\/latex], then the inverse function [latex]f^{-1}\\left(x\\right)[\/latex] must contain the point [latex]\\left(b, a\\right)[\/latex].<\/section>\n
            We begin with the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]. Because every logarithmic function of this form is the inverse of an exponential function of the form [latex]y={b}^{x}[\/latex], their graphs will be reflections of each other across the line [latex]y=x[\/latex].<\/section>\n
            \n
            To illustrate this, we can observe the relationship between the input and output values of [latex]y={2}^{x}[\/latex] and its equivalent logarithmic form [latex]x={\\mathrm{log}}_{2}\\left(y\\right)[\/latex] in the table below.<\/section>\n
            \n\n\n\n\n\n
            [latex]x[\/latex]<\/strong><\/td>\n[latex]\u20133[\/latex]<\/td>\n[latex]\u20132[\/latex]<\/td>\n[latex]\u20131[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n[latex]3[\/latex]<\/td>\n<\/tr>\n
            [latex]{2}^{x}=y[\/latex]<\/strong><\/td>\n[latex]\\frac{1}{8}[\/latex]<\/td>\n[latex]\\frac{1}{4}[\/latex]<\/td>\n[latex]\\frac{1}{2}[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n[latex]8[\/latex]<\/td>\n<\/tr>\n
            [latex]{\\mathrm{log}}_{2}\\left(y\\right)=x[\/latex]<\/strong><\/td>\n[latex]\u20133[\/latex]<\/td>\n[latex]\u20132[\/latex]<\/td>\n[latex]\u20131[\/latex]<\/td>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

            Using the inputs and outputs from the table above, we can build another table to observe the relationship between points on the graphs of the inverse functions [latex]f\\left(x\\right)={2}^{x}[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/p>\n\n\n\n\n
            [latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n[latex]\\left(-3,\\frac{1}{8}\\right)[\/latex]<\/td>\n[latex]\\left(-2,\\frac{1}{4}\\right)[\/latex]<\/td>\n[latex]\\left(-1,\\frac{1}{2}\\right)[\/latex]<\/td>\n[latex]\\left(0,1\\right)[\/latex]<\/td>\n[latex]\\left(1,2\\right)[\/latex]<\/td>\n[latex]\\left(2,4\\right)[\/latex]<\/td>\n[latex]\\left(3,8\\right)[\/latex]<\/td>\n<\/tr>\n
            [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex]<\/strong><\/td>\n[latex]\\left(\\frac{1}{8},-3\\right)[\/latex]<\/td>\n[latex]\\left(\\frac{1}{4},-2\\right)[\/latex]<\/td>\n[latex]\\left(\\frac{1}{2},-1\\right)[\/latex]<\/td>\n[latex]\\left(1,0\\right)[\/latex]<\/td>\n[latex]\\left(2,1\\right)[\/latex]<\/td>\n[latex]\\left(4,2\\right)[\/latex]<\/td>\n[latex]\\left(8,3\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

            As we would expect, the [latex]x[\/latex]and [latex]y[\/latex]-coordinates are reversed for the inverse functions. The figure below\u00a0shows the graphs of [latex]f[\/latex]\u00a0and [latex]g[\/latex].<\/p>\n

            \"Graph
            Notice that the graphs of [latex]f\\left(x\\right)={2}^{x}[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] are reflections about the line y = x since they are inverses of each other.<\/figcaption><\/figure>\n

            Observe the following from the graph:<\/p>\n

              \n
            • [latex]f\\left(x\\right)={2}^{x}[\/latex] has a y<\/em>-intercept at [latex]\\left(0,1\\right)[\/latex] and [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex] has an x<\/em>-intercept at [latex]\\left(1,0\\right)[\/latex].<\/li>\n
            • The domain of [latex]f\\left(x\\right)={2}^{x}[\/latex], [latex]\\left(-\\infty ,\\infty \\right)[\/latex], is the same as the range of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/li>\n
            • The range of [latex]f\\left(x\\right)={2}^{x}[\/latex], [latex]\\left(0,\\infty \\right)[\/latex], is the same as the domain of [latex]g\\left(x\\right)={\\mathrm{log}}_{2}\\left(x\\right)[\/latex].<\/li>\n<\/ul>\n<\/section>\n<\/section>\n
              \n

              characteristics of the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\n

              For any real number [latex]x[\/latex] and constant [latex]b \\gt 0[\/latex], [latex]b\\ne 1[\/latex], we can see the following characteristics in the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]:<\/p>\n

                \n
              • one-to-one function<\/li>\n
              • vertical asymptote: [latex]x\u00a0= 0[\/latex]<\/li>\n
              • domain: [latex]\\left(0,\\infty \\right)[\/latex]<\/li>\n
              • range: [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/li>\n
              • [latex]x[\/latex]–<\/em>intercept: [latex]\\left(1,0\\right)[\/latex] and key point [latex]\\left(b,1\\right)[\/latex]<\/li>\n
              • [latex]y[\/latex]-intercept: none<\/li>\n
              • increasing if [latex]b \\gt 1[\/latex]<\/li>\n
              • decreasing if [latex]0 \\lt b \\lt 1[\/latex]<\/li>\n<\/ul>\n

                 <\/p>\n

                \"\"<\/p>\n<\/section>\n

                How To: Given a logarithmic function Of the form [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph the function<\/strong><\/p>\n
                  \n
                1. Draw and label the vertical asymptote, [latex]x = 0[\/latex].<\/li>\n
                2. Plot the [latex]x[\/latex]–<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/li>\n
                3. Plot the key point [latex]\\left(b,1\\right)[\/latex].<\/li>\n
                4. Draw a smooth curve through the points.<\/li>\n
                5. State the domain, [latex]\\left(0,\\infty \\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote, [latex]x = 0[\/latex].<\/li>\n<\/ol>\n<\/section>\n
                  Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\n