{"id":1060,"date":"2025-07-22T00:08:14","date_gmt":"2025-07-22T00:08:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1060"},"modified":"2026-03-17T21:01:38","modified_gmt":"2026-03-17T21:01:38","slug":"graphs-of-logarithms-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-logarithms-learn-it-3\/","title":{"raw":"Graphs of Logarithms: Learn It 3","rendered":"Graphs of Logarithms: Learn It 3"},"content":{"raw":"

Graphing Transformations of Logarithmic Functions<\/h2>\r\nTransformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] without loss of shape.\r\n

Graphing a Horizontal Shift of\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nWhen a constant [latex]c[\/latex]\u00a0is added to the input of the parent function [latex]f\\left(x\\right)=\\text{log}_{b}\\left(x\\right)[\/latex], the result is a horizontal shift<\/strong> [latex]c[\/latex]\u00a0units in the opposite<\/em> direction of the sign on [latex]c[\/latex].\r\n\r\nTo visualize horizontal shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift left,\u00a0[latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], and the shift right, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex] where\u00a0[latex]c\u00a0> 0[\/latex].\r\n\r\n
Using an online graphing calculator, plot the functions\u00a0[latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] and\u00a0[latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex]Investigate the following questions:\r\n
    \r\n \t
  • Adjust the [latex]c[\/latex] value to [latex]4[\/latex].<\/li>\r\n \t
  • Which direction does the graph of [latex]g(x)[\/latex] shift? What is the vertical asymptote, [latex]x[\/latex]-intercept, and equation for this new function? How do the domain and range change?<\/li>\r\n \t
  • Which direction does the graph of [latex]h(x)[\/latex] shift? What is the vertical asymptote, [latex]x[\/latex]-intercept, and equation for this new function? How do the domain and range change?<\/li>\r\n<\/ul>\r\n<\/section>The graphs below summarize the changes in the [latex]x[\/latex]-intercepts, vertical asymptotes, and equations\u00a0of a logarithmic function that has been shifted either right or left.\r\n\r\n\"Graph\r\n\r\n
    \r\n

    horizontal shifts of the parent function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nFor any constant [latex]c[\/latex], the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex]\r\n
      \r\n \t
    • shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left [latex]c[\/latex]\u00a0units if [latex]c\u00a0> 0[\/latex].<\/li>\r\n \t
    • shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right [latex]c[\/latex]\u00a0units if [latex]c\u00a0< 0[\/latex].<\/li>\r\n \t
    • has the vertical asymptote [latex]x = \u2013c[\/latex].<\/li>\r\n \t
    • has domain [latex]\\left(-c,\\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)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], graph the Horizontal Shift<\/strong>\r\n
        \r\n \t
      1. Identify the horizontal shift:\r\n
          \r\n \t
        • If [latex]c > 0[\/latex], shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left [latex]c[\/latex]\u00a0units.<\/li>\r\n \t
        • If [latex]c\u00a0< 0[\/latex], shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right [latex]c[\/latex]\u00a0units.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
        • Draw the vertical asymptote [latex]x\u00a0= \u2013c[\/latex].<\/li>\r\n \t
        • Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting [latex]c[\/latex]\u00a0from the\u00a0[latex]x[\/latex]\u00a0coordinate in each point.<\/li>\r\n \t
        • Label the three points.<\/li>\r\n \t
        • The domain is [latex]\\left(-c,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is [latex]x\u00a0<\/em>= \u2013c[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
          Graph the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.[reveal-answer q=\"368750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368750\"]Parent Function: [latex]y = \\mathrm{log}_{3}(x)[\/latex]<\/strong>\r\n
            \r\n \t
          • Consider the three key points from the parent function: [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].<\/li>\r\n<\/ul>\r\n\"GraphSince the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex], we notice [latex]x+\\left(-2\\right)=x - 2[\/latex].\r\n
              \r\n \t
            • Thus [latex]c\u00a0= \u20132[\/latex], so [latex]c \\lt 0[\/latex]. This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] right<\/strong> [latex]2[\/latex] units.<\/li>\r\n \t
            • The vertical asymptote<\/strong> is [latex]x=-\\left(-2\\right)[\/latex] or [latex]x = 2[\/latex].<\/li>\r\n \t
            • The new coordinates are found by adding [latex]2[\/latex]<\/span> to the <\/span>[latex]x[\/latex] coordinates of each point from the parent function. <\/span>Therefore, the points are [latex]\\left(\\frac{7}{3},-1\\right)[\/latex], [latex]\\left(3,0\\right)[\/latex], and [latex]\\left(5,1\\right)[\/latex].<\/li>\r\n<\/ul>\r\nCharacteristics of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex]:\r\n
                \r\n \t
              • The domain<\/strong> is [latex]\\left(2,\\infty \\right)[\/latex]<\/li>\r\n \t
              • The range<\/strong> is [latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/li>\r\n \t
              • The vertical asymptote is [latex]x= 2[\/latex].<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>
                [ohm_question hide_question_numbers=1]321446[\/ohm_question]<\/section>
                [ohm_question hide_question_numbers=1]321447[\/ohm_question]<\/section>\r\n

                Graphing a Vertical Shift of\u00a0[latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nWhen a constant [latex]d[\/latex]\u00a0is added to the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex], the result is a vertical shift [latex]d[\/latex]\u00a0units in the direction of the sign of\u00a0[latex]d[\/latex]. To visualize vertical shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift up, [latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex], and the shift down, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)-d[\/latex].\r\n\r\n\"Graph\r\n\r\n
                \r\n

                vertical shifts of the parent function [latex]y=\\text{log}_{b}\\left(x\\right)[\/latex]<\/h3>\r\nFor any constant [latex]d[\/latex], the function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex]\r\n
                  \r\n \t
                • shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up [latex]d[\/latex]\u00a0units if [latex]d\u00a0> 0[\/latex].<\/li>\r\n \t
                • shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down [latex]d[\/latex]\u00a0units if [latex]d\u00a0< 0[\/latex].<\/li>\r\n \t
                • has the vertical asymptote [latex]x\u00a0= 0[\/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)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex], graph the Vertical Shift<\/strong>\r\n
                    \r\n \t
                  1. Identify the vertical shift:\r\n
                      \r\n \t
                    • If [latex]d\u00a0> 0[\/latex], shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] up [latex]d[\/latex]\u00a0units.<\/li>\r\n \t
                    • If [latex]d\u00a0< 0[\/latex], shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] down [latex]d[\/latex]units.<\/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 adding [latex]d[\/latex]\u00a0to the [latex]y[\/latex]coordinate of each point.<\/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\u00a0= 0[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
                      Sketch a graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.[reveal-answer q=\"43912\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"43912\"][caption id=\"\" align=\"alignright\" width=\"326\"]\"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]Parent Function: [latex]y = \\mathrm{log}_{3}(x)[\/latex]<\/strong>\r\n
                        \r\n \t
                      • Consider the three key points from the parent function: [latex]\\left(\\frac{1}{3},-1\\right)[\/latex], [latex]\\left(1,0\\right)[\/latex], and [latex]\\left(3,1\\right)[\/latex].<\/li>\r\n<\/ul>\r\nSince the function is [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)-2[\/latex], we notice d\u00a0<\/em>= \u20132. Thus d\u00a0<\/em>< 0.\r\n
                          \r\n \t
                        • This means we will shift the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x\\right)[\/latex] down [latex]2[\/latex] units.<\/li>\r\n \t
                        • The vertical asymptote is [latex]x = 0[\/latex].<\/li>\r\n \t
                        • The new coordinates are found by subtracting [latex]2[\/latex] from the [latex]y[\/latex] coordinates of each point from the parent function. T<\/span>he points are [latex]\\left(\\frac{1}{3},-3\\right)[\/latex], [latex]\\left(1,-2\\right)[\/latex], and [latex]\\left(3,-1\\right)[\/latex].<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/section>
                          [ohm_question hide_question_numbers=1]321448[\/ohm_question]<\/section>
                          [ohm_question hide_question_numbers=1]321449[\/ohm_question]<\/section>","rendered":"

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

                          Transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] without loss of shape.<\/p>\n

                          Graphing a Horizontal Shift of\u00a0[latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/h3>\n

                          When a constant [latex]c[\/latex]\u00a0is added to the input of the parent function [latex]f\\left(x\\right)=\\text{log}_{b}\\left(x\\right)[\/latex], the result is a horizontal shift<\/strong> [latex]c[\/latex]\u00a0units in the opposite<\/em> direction of the sign on [latex]c[\/latex].<\/p>\n

                          To visualize horizontal shifts, we can observe the general graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] alongside the shift left,\u00a0[latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], and the shift right, [latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex] where\u00a0[latex]c\u00a0> 0[\/latex].<\/p>\n

                          Using an online graphing calculator, plot the functions\u00a0[latex]g\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] and\u00a0[latex]h\\left(x\\right)={\\mathrm{log}}_{b}\\left(x-c\\right)[\/latex]Investigate the following questions:<\/p>\n
                            \n
                          • Adjust the [latex]c[\/latex] value to [latex]4[\/latex].<\/li>\n
                          • Which direction does the graph of [latex]g(x)[\/latex] shift? What is the vertical asymptote, [latex]x[\/latex]-intercept, and equation for this new function? How do the domain and range change?<\/li>\n
                          • Which direction does the graph of [latex]h(x)[\/latex] shift? What is the vertical asymptote, [latex]x[\/latex]-intercept, and equation for this new function? How do the domain and range change?<\/li>\n<\/ul>\n<\/section>\n

                            The graphs below summarize the changes in the [latex]x[\/latex]-intercepts, vertical asymptotes, and equations\u00a0of a logarithmic function that has been shifted either right or left.<\/p>\n

                            \"Graph<\/p>\n

                            \n

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

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

                              \n
                            • shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left [latex]c[\/latex]\u00a0units if [latex]c\u00a0> 0[\/latex].<\/li>\n
                            • shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right [latex]c[\/latex]\u00a0units if [latex]c\u00a0< 0[\/latex].<\/li>\n
                            • has the vertical asymptote [latex]x = \u2013c[\/latex].<\/li>\n
                            • has domain [latex]\\left(-c,\\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)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex], graph the Horizontal Shift<\/strong><\/p>\n
                                \n
                              1. Identify the horizontal shift:\n
                                  \n
                                • If [latex]c > 0[\/latex], shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] left [latex]c[\/latex]\u00a0units.<\/li>\n
                                • If [latex]c\u00a0< 0[\/latex], shift the graph of [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] right [latex]c[\/latex]\u00a0units.<\/li>\n<\/ul>\n<\/li>\n
                                • Draw the vertical asymptote [latex]x\u00a0= \u2013c[\/latex].<\/li>\n
                                • Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting [latex]c[\/latex]\u00a0from the\u00a0[latex]x[\/latex]\u00a0coordinate in each point.<\/li>\n
                                • Label the three points.<\/li>\n
                                • The domain is [latex]\\left(-c,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is [latex]x\u00a0<\/em>= \u2013c[\/latex].<\/li>\n<\/ol>\n<\/section>\n
                                  Graph the function [latex]f\\left(x\\right)={\\mathrm{log}}_{3}\\left(x - 2\\right)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n