{"id":1059,"date":"2025-07-22T00:08:18","date_gmt":"2025-07-22T00:08:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1059"},"modified":"2026-03-17T22:46:04","modified_gmt":"2026-03-17T22:46:04","slug":"graphs-of-logarithms-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-logarithms-learn-it-4\/","title":{"raw":"Graphs of Logarithms: Learn It 4","rendered":"Graphs of Logarithms: Learn It 4"},"content":{"raw":"

Graphing Transformations of Logarithmic Functions Cont.<\/h2>\r\n

Graphing Stretches and Compressions of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nWhen the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by a constant [latex]a > 0[\/latex], the result is a vertical stretch<\/strong> or compression<\/strong> of the original graph. To visualize stretches and compressions, we set [latex]a\u00a0> 1[\/latex] and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the vertical stretch, [latex]g\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], and the vertical compression, [latex]h\\left(x\\right)=\\frac{1}{a}{\\mathrm{log}}_{b}\\left(x\\right)[\/latex].\r\n\r\n
Using an online graphing calculator plot the functions [latex]g(x) = a\\log_{b}{x}[\/latex]\u00a0 and\u00a0 [latex]h(x) = \\frac{1}{a}\\log_{b}{x}[\/latex]. One represents a vertical compression of the other. You may select any value for [latex]b[\/latex], though one between [latex]2[\/latex] and [latex]5[\/latex] will be easier to see. Experiment with various [latex]a[\/latex] values between [latex]1[\/latex] and [latex]10[\/latex]. As you investigate, consider the following questions:\r\n
    \r\n \t
  • Both the vertical stretch and compression produce graphs that are increasing. Which transformation produces a function that increases faster?<\/li>\r\n \t
  • One of the key points that is commonly defined for transformations of a logarithmic function comes from finding the input that gives an output of [latex]y = 1[\/latex]. This point can help you determine whether a graph is the result of a vertical compression or stretch. Explain why.<\/li>\r\n<\/ul>\r\n<\/section>The graphs below summarize the key features of the resulting graphs of vertical stretches and compressions of logarithmic functions.\r\n\r\n\"Graph\r\n\r\n
    \r\n

    vertical stretches and compressions of the parent function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nFor any constant [latex]a > 1[\/latex], the function [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\r\n
      \r\n \t
    • stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex]\u00a0if [latex]a\u00a0> 1[\/latex].<\/li>\r\n \t
    • compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex]\u00a0if [latex]0 < a\u00a0< 1[\/latex].<\/li>\r\n \t
    • has the vertical asymptote [latex]x\u00a0= 0[\/latex].<\/li>\r\n \t
    • has the [latex]x[\/latex]-intercept [latex]\\left(1,0\\right)[\/latex].<\/li>\r\n \t
    • has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\r\n \t
    • has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>
      How To: Given a logarithmic function Of the form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], [latex]a>0[\/latex], graph the Stretch or Compression<\/strong>\r\n
        \r\n \t
      1. Identify the vertical stretch or compression:\r\n
          \r\n \t
        • If [latex]|a|>1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is stretched by a factor of [latex]a[\/latex]\u00a0units.<\/li>\r\n \t
        • If [latex]|a|<1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is compressed by a factor of [latex]a[\/latex]\u00a0units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
        • Draw the vertical asymptote [latex]x\u00a0= 0[\/latex].<\/li>\r\n \t
        • Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the [latex]y[\/latex]\u00a0coordinates in each point by [latex]a[\/latex].<\/li>\r\n \t
        • Label the three points.<\/li>\r\n \t
        • The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is [latex]x = 0[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
          Graph the function [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.[reveal-answer q=\"595868\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"595868\"][caption id=\"\" align=\"alignright\" width=\"375\"]\"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\u00a0= 0.[\/caption]Parent Function: [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex]<\/strong>\r\n
            \r\n \t
          • Consider the three key points from the parent function, [latex]\\left(\\frac{1}{4},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(4,1\\right)[\/latex].<\/li>\r\n<\/ul>\r\nSince the function is [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex], we will note that [latex]a = 2[\/latex].\r\n
              \r\n \t
            • This means we will stretch the function [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(x\\right)[\/latex] by a factor of [latex]2[\/latex].<\/li>\r\n \t
            • The vertical asymptote is [latex]x\u00a0= 0[\/latex].<\/li>\r\n \t
            • The new coordinates are found by multiplying the [latex]y[\/latex]\u00a0coordinates of each point by [latex]2[\/latex]. Thus, the transformed\u00a0points are [latex]\\left(\\frac{1}{4},-2\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(4,\\text{2}\\right)[\/latex].<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>
              Sketch the graph of [latex]f\\left(x\\right)=5\\mathrm{log}\\left(x+2\\right)[\/latex]. State the domain, range, and asymptote.[reveal-answer q=\"804029\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"804029\"]Remember, what happens inside parentheses happens first. First, we move the graph left [latex]2[\/latex] units and then stretch the function vertically by a factor of [latex]5[\/latex]. The vertical asymptote will be shifted to [latex]x\u00a0= \u20132[\/latex]. The [latex]x[\/latex]-intercept will be [latex]\\left(-1,0\\right)[\/latex]. The domain will be [latex]\\left(-2,\\infty \\right)[\/latex]. Two points will help give the shape of the graph: [latex]\\left(-1,0\\right)[\/latex] and [latex]\\left(8,5\\right)[\/latex]. We chose [latex]x\u00a0= 8[\/latex] as the [latex]x[\/latex]-coordinate of one point to graph because when [latex]x\u00a0= 8[\/latex], [latex]x\u00a0+ 2 = 10[\/latex], the base of the common logarithm.[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph The domain is [latex]\\left(-2,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x\u00a0= \u20132.[\/caption][\/hidden-answer]<\/section>\r\n

              Graphing Reflections of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nWhen the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by [latex]\u20131[\/latex], the result is a reflection<\/strong> about the [latex]x[\/latex]-axis. When the input<\/em> is multiplied by [latex]\u20131[\/latex], the result is a reflection about the [latex]y[\/latex]-axis. To visualize reflections, we restrict [latex]b\u00a0> 1[\/latex] and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the reflection about the [latex]x[\/latex]-axis, [latex]g\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex], and the reflection about the [latex]y[\/latex]-axis, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex].\r\n\r\n
              Using an online graphing calculator, plot the functions [latex]f(x) = \\log_{b}{x},\\text{ }g(x)=-\\log_{b}{x},\\text{ and }h(x) = \\log_{b}({-x}) [\/latex]. You may select any value for [latex]b[\/latex], though one between [latex]2[\/latex] and [latex]5[\/latex] will be easier to see. Also plot the point [latex](b,1)[\/latex]. Consider the following questions:\r\n
                \r\n \t
              • Which graph, [latex]g(x) = -\\log_{b}{x} \\text{ or }h(x) = \\log_{b}({-x})[\/latex] represents a vertical reflection? \u00a0Which one represents a horizontal reflection?<\/li>\r\n \t
              • You already added the point [latex](b,1)[\/latex] as a point of interest for the function [latex]f(x)[\/latex]. Using the variable [latex]b[\/latex] as your [latex]x[\/latex] value, add the corresponding points of interest for [latex]g(x)\\text{ and }h(x)[\/latex].<\/li>\r\n \t
              • Does the vertical asymptote change when you reflect the graph of [latex]f(x)[\/latex] either vertically or horizontally?<\/li>\r\n<\/ul>\r\n<\/section>The graphs below summarize the key characteristics of reflecting [latex]f(x) = \\log_{b}{x}[\/latex] horizontally and vertically.\r\n\r\n\"Graph\r\n\r\n
                \r\n

                reflections of the parent function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nThe function [latex]f\\left(x\\right)={\\mathrm{-log}}_{b}\\left(x\\right)[\/latex]\r\n
                  \r\n \t
                • reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the [latex]x[\/latex]-axis.<\/li>\r\n \t
                • has domain [latex]\\left(0,\\infty \\right)[\/latex], range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and vertical asymptote\u00a0[latex]x\u00a0= 0[\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\nThe function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]\r\n
                    \r\n \t
                  • reflects the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the [latex]y[\/latex]-axis.<\/li>\r\n \t
                  • has domain [latex]\\left(-\\infty ,0\\right)[\/latex].<\/li>\r\n \t
                  • has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex] and vertical asymptote\u00a0[latex]x\u00a0= 0[\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/section>
                    How To: Given a logarithmic function with the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], graph a Reflection<\/strong>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                    [latex]\\text{If }f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/th>\r\n[latex]\\text{If }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
                    1. Draw the vertical asymptote, [latex]x\u00a0= 0[\/latex].<\/td>\r\n1. Draw the vertical asymptote, [latex]x\u00a0= 0[\/latex].<\/td>\r\n<\/tr>\r\n
                    2. Plot the x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\r\n2. Plot the x-<\/em>intercept, [latex]\\left(1,0\\right)[\/latex].<\/td>\r\n<\/tr>\r\n
                    3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the [latex]x[\/latex]-axis.<\/td>\r\n3. Reflect the graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] about the [latex]y[\/latex]-axis.<\/td>\r\n<\/tr>\r\n
                    4. Draw a smooth curve through the points.<\/td>\r\n4. Draw a smooth curve through the points.<\/td>\r\n<\/tr>\r\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\u00a0= 0[\/latex].<\/td>\r\n5. State the domain, [latex]\\left(-\\infty ,0\\right)[\/latex], the range, [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote [latex]x\u00a0= 0[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>
                    Graph the function [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.[reveal-answer q=\"843271\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843271\"][caption id=\"\" align=\"alignright\" width=\"400\"]\"Graph The domain is [latex]\\left(-\\infty ,0\\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x\u00a0= 0.[\/caption]Before graphing [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex], identify the behavior and key points for the graph.\r\n
                      \r\n \t
                    • Since [latex]b\u00a0= 10[\/latex] is greater than one, we know that the parent function is increasing. Since the input<\/em> value is multiplied by [latex]\u20131[\/latex], [latex]f[\/latex]\u00a0is a reflection of the parent graph about the [latex]y[\/latex]-<\/em>axis. Thus, [latex]f\\left(x\\right)=\\mathrm{log}\\left(-x\\right)[\/latex] will be decreasing as x<\/em>\u00a0moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote [latex]x\u00a0= 0[\/latex].<\/li>\r\n \t
                    • The [latex]x[\/latex]-intercept is [latex]\\left(-1,0\\right)[\/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\n[\/hidden-answer]\r\n\r\n<\/section>
                      [ohm_question hide_question_numbers=1]321451[\/ohm_question]<\/section>
                      [ohm_question hide_question_numbers=1]321452[\/ohm_question]<\/section>","rendered":"

                      Graphing Transformations of Logarithmic Functions Cont.<\/h2>\n

                      Graphing Stretches and Compressions of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n

                      When the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by a constant [latex]a > 0[\/latex], the result is a vertical stretch<\/strong> or compression<\/strong> of the original graph. To visualize stretches and compressions, we set [latex]a\u00a0> 1[\/latex] and observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the vertical stretch, [latex]g\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], and the vertical compression, [latex]h\\left(x\\right)=\\frac{1}{a}{\\mathrm{log}}_{b}\\left(x\\right)[\/latex].<\/p>\n

                      Using an online graphing calculator plot the functions [latex]g(x) = a\\log_{b}{x}[\/latex]\u00a0 and\u00a0 [latex]h(x) = \\frac{1}{a}\\log_{b}{x}[\/latex]. One represents a vertical compression of the other. You may select any value for [latex]b[\/latex], though one between [latex]2[\/latex] and [latex]5[\/latex] will be easier to see. Experiment with various [latex]a[\/latex] values between [latex]1[\/latex] and [latex]10[\/latex]. As you investigate, consider the following questions:<\/p>\n
                        \n
                      • Both the vertical stretch and compression produce graphs that are increasing. Which transformation produces a function that increases faster?<\/li>\n
                      • One of the key points that is commonly defined for transformations of a logarithmic function comes from finding the input that gives an output of [latex]y = 1[\/latex]. This point can help you determine whether a graph is the result of a vertical compression or stretch. Explain why.<\/li>\n<\/ul>\n<\/section>\n

                        The graphs below summarize the key features of the resulting graphs of vertical stretches and compressions of logarithmic functions.<\/p>\n

                        \"Graph<\/p>\n

                        \n

                        vertical stretches and compressions of the parent function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\n

                        For any constant [latex]a > 1[\/latex], the function [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/p>\n

                          \n
                        • stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex]\u00a0if [latex]a\u00a0> 1[\/latex].<\/li>\n
                        • compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of [latex]a[\/latex]\u00a0if [latex]0 < a\u00a0< 1[\/latex].<\/li>\n
                        • has the vertical asymptote [latex]x\u00a0= 0[\/latex].<\/li>\n
                        • has the [latex]x[\/latex]-intercept [latex]\\left(1,0\\right)[\/latex].<\/li>\n
                        • has domain [latex]\\left(0,\\infty \\right)[\/latex].<\/li>\n
                        • has range [latex]\\left(-\\infty ,\\infty \\right)[\/latex].<\/li>\n<\/ul>\n<\/section>\n
                          How To: Given a logarithmic function Of the form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex], [latex]a>0[\/latex], graph the Stretch or Compression<\/strong><\/p>\n
                            \n
                          1. Identify the vertical stretch or compression:\n
                              \n
                            • If [latex]|a|>1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is stretched by a factor of [latex]a[\/latex]\u00a0units.<\/li>\n
                            • If [latex]|a|<1[\/latex], the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is compressed by a factor of [latex]a[\/latex]\u00a0units.<\/li>\n<\/ul>\n<\/li>\n
                            • Draw the vertical asymptote [latex]x\u00a0= 0[\/latex].<\/li>\n
                            • Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the [latex]y[\/latex]\u00a0coordinates in each point by [latex]a[\/latex].<\/li>\n
                            • Label the three points.<\/li>\n
                            • The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is [latex]x = 0[\/latex].<\/li>\n<\/ol>\n<\/section>\n
                              Graph the function [latex]f\\left(x\\right)=2{\\mathrm{log}}_{4}\\left(x\\right)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n