468 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions More general logarithmic functions can be obtained by forming the composition of ƒ1x2 = loga x with a function g1x2. For example, if ƒ1x2 = log2 x and g1x2 = x - 1, then 1ƒ∘ g21x2 = ƒ1g1x22 = log2 1x - 12. The next example shows how to graph such functions. CAUTION If we write a logarithmic function in exponential form in order to graph it, as in Example 3(b), we start with y-values to calculate corresponding x-values. Be careful to write the values in the ordered pairs in the correct order. EXAMPLE 4 GraphingTranslated Logarithmic Functions Graph each function. Give the domain and range. (a) ƒ1x2 = log2 1x - 12 (b) ƒ1x2 = 1log3 x2 - 1 (c) ƒ1x2 = log4 1x + 22 + 1 SOLUTION (a) The graph of ƒ1x2 = log2 1x - 12 is the graph of g1x2 = log2 x translated to the right 1 unit. The vertical asymptote has equation x = 1. Because logarithms can be found only for positive numbers, we solve x - 1 70 to find the domain, 11, ∞2. To determine ordered pairs to plot, use the equivalent exponential form of the equation y = log2 1x - 12. y = log2 1x - 12 x - 1 = 2y Write in exponential form. x = 2y + 1 Add 1. We first choose values for y and then calculate each of the corresponding x-values. The range is 1-∞, ∞2. See Figure 30. f(x) = log2 (x – 1) x = 1 (3, 1) (2, 0) (5, 2) x 4 2 –2 4 6 8 y 0 Figure 30 x 3 6 9 f(x) = (log3 x) – 1 0 (1, 21) (3, 0) (9, 1) 2 –2 4 y Figure 31 (b) The function ƒ1x2 = 1log3 x2 - 1 has the same graph as g1x2 = log3 x translated down 1 unit. We find ordered pairs to plot by writing the equation y = 1log3 x2 - 1 in exponential form. y = 1log3 x2 - 1 y + 1 = log3 x Add 1. x = 3y+1 Write in exponential form. Again, choose y-values and calculate the corresponding x-values. The graph is shown in Figure 31. The domain is 10, ∞2, and the range is 1-∞, ∞2.
RkJQdWJsaXNoZXIy NjM5ODQ=