b.Because is the inverse function of the graph of gisobtained by plotting the points and connecting them with a smoothcurve. The graph of is a reflection of the graph of fin the line asshown in Figure 3.18.Now try Exercise 43.Before you can confirm the result of Example 4 using a graphing utility, youneed to know how to enter You will learn how to do this using the change-of-base formuladiscussed in Section 3.3.log2 x.yx,gf x, xf x2x,g xlog2 xf x2x,g xlog2xf x2xSTUDY TIPIn Example 5, you can alsosketch the graph of by evaluating the inverse function of f,forseveral values of x. Plot thepoints, sketch the graph of g, andthen reflect the graph in the lineto obtain the graph of f.yxg x10x,f xlog10 xExample 5Sketching the Graph of a Logarithmic FunctionSketch the graph of the common logarithmic function by hand.SolutionBegin by constructing a table of values. Note that some of the values can beobtained without a calculator by using the Inverse Property of Logarithms. Othersrequire a calculator. Next, plot the points and connect them with a smooth curve,as shown in Figure 3.19.Now try Exercise 47.The nature of the graph in Figure 3.19 is typical of functions of the formThey have one x-intercept and one vertical asymptote.Notice how slowly the graph rises for x>1.f xlogax, a>1.f xlog10xx210123f x2x14121248Without calculatorWith calculator110258010.3010.6990.90312f xlog10x1101100xFigure 3.19Figure 3.18

Section 3.2Logarithmic Functions and Their Graphs199Example 6Transformations of Graphs of Logarithmic FunctionsEach of the following functions is a transformation of the graph of a.Because the graph of can be obtained byshifting the graph ofone unit to the right, as shown in Figure 3.20.b.Because the graph of can be obtained byshifting the graph oftwo units upward, as shown in Figure 3.21.Figure 3.20Figure 3.21Notice that the transformation in Figure 3.21 keeps the y-axis as a vertical asymp-tote, but the transformation in Figure 3.20 yields the new vertical asymptoteNow try Exercise 57.x1.-15-1(1, 2)(1, 0)3f(x) = log10xh(x) = 2 + log10x(2, 0)(1, 0)−0.54−21x= 1f(x) = log10xg(x) = log10(x−1)fhh x2log10x2f x,fgg xlog10x1f x1 ,f xlog10x.Library of Parent Functions: Logarithmic FunctionThe logarithmic functionis the inverse function of the exponential function. Its domain is the set ofpositive real numbers and its range is the set of all real numbers. This is theopposite of the exponential function. Moreover, the logarithmic function hasthe y-axis as a vertical asymptote, whereas the exponential function has thex-axis as a horizontal asymptote. Many real-life phenomena with a slow rateof growth can be modeled by logarithmic functions. The basic characteris-tics of the logarithmic function are summarized below. A review of logarith-mic functions can be found in the Study Capsules.

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- Logarithmic Functions, Natural logarithm, Logarithm, Inverse Property