463 4.3 Logarithmic Functions In calculus, the following can be shown. ex = 1 + x + x2 2 # 1 + x3 3 # 2 # 1 + x4 4 # 3 # 2 # 1 + x5 5 # 4 # 3 # 2 # 1 + g Using more terms, one can obtain a more accurate approximation for ex. 125. Use the terms shown, and replace x with 1 to approximate e1 = e to three decimal places. Check the result with a calculator. 126. Use the terms shown, and replace x with -0.05 to approximate e-0.05 to four decimal places. Check the result with a calculator. Relating Concepts For individual or collaborative investigation (Exercises 127–132) Consider ƒ1x2 = ax, where a 71. Work these exercises in order. 127. Is ƒ a one-to-one function? If so, what kind of related function exists for ƒ? 128. If ƒ has an inverse function ƒ -1, sketch ƒ and ƒ -1 on the same set of axes. 129. If ƒ -1 exists, find an equation for y = ƒ -11x2. (You need not solve for y.) 130. If a = 10, what is the equation for y = ƒ -11x2? (You need not solve for y.) 131. If a = e, what is the equation for y = ƒ -11x2? (You need not solve for y.) 132. If the point 1p, q2 is on the graph of ƒ, then the point is on the graph of ƒ -1. 4.3 Logarithmic Functions ■ Logarithms ■ Logarithmic Equations ■ Logarithmic Functions ■ Properties of Logarithms Logarithms The previous section dealt with exponential functions of the form y = ax for all positive values of a, where a≠1. The horizontal line test shows that exponential functions are one-to-one and thus have inverse functions. The equation defining the inverse of a function is found by interchanging x and y in the equation that defines the function. Starting with y = ax and interchanging x and y yields x = ay. Here y is the exponent to which a must be raised in order to obtain x. We call this exponent a logarithm, symbolized by the abbreviation “log.” The expression log a x represents the logarithm in this discussion. The number a is the base of the logarithm, and x is the argument of the expression. It is read “logarithm with base a of x,” or “logarithm of x with base a,” or “base a logarithm of x.” Logarithm For all real numbers y and all positive numbers a and x, where a ≠1, y =log a x is equivalent to x =a y. The expression log a x represents the exponent to which the base a must be raised in order to obtain x.
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