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>\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\n | As we\u2019d expect, the x<\/em>- and y<\/em>-coordinates are reversed for the inverse functions. The figure below\u00a0shows the graph of f<\/em>\u00a0and g<\/em>.<\/p>\r\n\r\n Figure 2.\u00a0<\/strong>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\u00a0<\/em>= x<\/em>.<\/p>\r\n Observe the following from the graph:<\/p>\r\n\r\n For any real number x<\/em>\u00a0and constant b\u00a0<\/em>> 0, [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>\r\n\r\n Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\r\n\r\n<\/div>\r\n Before graphing, identify the behavior and key points for the graph.<\/p>\r\n\r\n Figure 5.\u00a0<\/strong>The domain is [latex]\\left(0,\\infty \\right)[\/latex], the range is [latex]\\left(-\\infty ,\\infty \\right)[\/latex], and the vertical asymptote is x<\/em> = 0.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n Graph [latex]f\\left(x\\right)={\\mathrm{log}}_{\\frac{1}{5}}\\left(x\\right)[\/latex]. State the domain, range, and asymptote.<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":" 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 its transformations: shifts, stretches, compressions, and reflections.<\/p>\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 with the form [latex]y={b}^{x}[\/latex], their graphs will be reflections of each other across the line [latex]y=x[\/latex]. To illustrate this, we can observe the relationship between the input and output values of [latex]y={2}^{x}[\/latex] and its equivalent [latex]x={\\mathrm{log}}_{2}\\left(y\\right)[\/latex] in the table below.<\/p>\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 As we\u2019d expect, the x<\/em>– and y<\/em>-coordinates are reversed for the inverse functions. The figure below\u00a0shows the graph of f<\/em>\u00a0and g<\/em>.<\/p>\n Figure 2.\u00a0<\/strong>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\u00a0<\/em>= x<\/em>.<\/p>\n Observe the following from the graph:<\/p>\n The video shows graphing logarithmic and exponential functions side by side.<\/p>\n |
![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010821\/CNX_Precalc_Figure_04_04_0022.jpg\")
<\/figure>\r\n![\"Two](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010821\/CNX_Precalc_Figure_04_04_003G2.jpg\")
Figure 3<\/b>[\/caption]<\/figure>\r\nFigure 3\u00a0shows how changing the base b<\/em>\u00a0in [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note:<\/em> recall that the function [latex]\\mathrm{ln}\\left(x\\right)[\/latex] has base [latex]e\\approx \\text{2}.\\text{718.)}[\/latex]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/CNX_Precalc_Figure_04_04_0042.jpg\")
Figure 4.\u00a0<\/strong>The graphs of three logarithmic functions with different bases, all greater than 1.[\/caption]\r\n\r\n<\/div>\r\n![\"Graph](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010822\/CNX_Precalc_Figure_04_04_0052.jpg\")
<\/span><\/figure>\r\n