518 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Concepts Examples Approximate log 0.045 and ln 247.1. log 0.045 ≈-1.3468 Use a calculator. ln 247.1 ≈5.5098 Approximate log8 7. log8 7 = log 7 log 8 = ln 7 ln 8 ≈0.9358 Use a calculator. Solve. e5x = 10 ln e5x = ln 10 Take natural logarithms. 5x = ln 10 ln e x = x, for all x. x = ln 10 5 Divide by 5. x ≈0.461 Use a calculator. The solution set can be written with the exact value, E ln 10 5 F, or with the approximate value, 50.4616. log2 1x 2 - 32 = log 2 6 x2 - 3 = 6 Property of logarithms x2 = 9 Add 3. x = {3 Take square roots. Both values check, so the solution set is 5{36. 4.4 Evaluating Logarithms and the Change-of-BaseTheorem 4.5 Exponential and Logarithmic Equations Property of Logarithms If x 70, y 70, a 70, and a≠1, then the following holds true. x =y is equivalent to log a x =log a y. 4.6 Applications and Models of Exponential Growth and Decay The formula for continuous compounding, A =Pe r t, is an example of exponential growth. Here, A is the compound amount if P dollars are invested at an annual interest rate r for t years. If P = $200, r = 3%, and t = 5 yr, find A. A = Pert A = 200e0.03152 Substitute. A ≈$232.37 Use a calculator. Common and Natural Logarithms For all positive numbers x, base 10 logarithms and base e logarithms are written as follows. log x =log10 x Common logarithm ln x =log e x Natural logarithm 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 Exponential Growth or Decay Function Let y0 be the amount or number present at time t = 0. Then, under certain conditions, the amount present at any time t is modeled by y =y0 ekt, where k is a constant.
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