482 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Proof Let y = loga x. Then ay = x Write in exponential form. logb ay = log b x Take the base b logarithm on each side. y logb a = logb x Power property y = logb x logb a Divide each side by logb a. loga x = logb x logb a . Substitute loga x for y. Any positive number other than 1 can be used for base b in the changeof-base theorem, but usually the only practical bases are e and 10 since most calculators give logarithms for these two bases only. Using the change-of-base theorem, we can graph an equation such as y = log2 x by directing the calculator to graph y = log x log 2 , or, equivalently, y = ln x ln 2 . 7 Change-of-Base Theorem For any positive real numbers x, a, and b, where a≠1 and b≠1, the following holds true. log a x = log b x log b a LOOKING AHEAD TO CALCULUS In calculus, natural logarithms are more convenient to work with than logarithms with other bases. The change-of-base theorem enables us to convert any logarithmic function to a natural logarithmic function. (a) (b) Figure 38 EXAMPLE 8 Using the Change-of-BaseTheorem Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. (a) log5 17 (b) log2 0.1 SOLUTION (a) We use natural logarithms to approximate this logarithm. Because log5 5 = 1 and log5 25 = 2, we can estimate log5 17 to be a number between 1 and 2. log5 17 = ln 17 ln 5 ≈1.7604 The first two entries in Figure 38(a) show that the results are the same whether natural or common logarithms are used. Check: 51.7604 ≈17
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