332 CHAPTER 5 Exponential and Logarithmic Functions Example 7 shows how to approximate a logarithm whose base is 2 by changing to logarithms involving the base e . In general, the Change-of-Base Formula is used. TECHNOLOGY NOTE Some calculators have features for evaluating logarithms with bases other than 10 or e . For example, the TI-84 Plus CE has the logBASE function (under Math > > Math A: logBASE). Consult the user’s manual for your calculator. j Graphing a Logarithmic Function Whose Base Is Neither 10 Nor e Use a graphing utility to graph = y x log . 2 EXAMPLE 9 THEOREM Change-of-Base Formula If ≠ ≠ a b 1, 1, and M are positive real numbers, then = M M a log log log a b b (9) Proof Let = y M log . a Then Property (7) Property (5) Solve for y . = y M loga ■ = = = = = a M a M y a M y M a M M a log log log log log log log log log y b y b b b b b a b b Because most calculators have keys for log and ln , in practice, the Change-of-Base Formula uses either = b 10 or = b e. That is, = = M M a M M a log log log and log ln ln a a Using the Change-of-Base Formula Approximate: (a) log 89 5 (b) log 5 2 Round answers to four decimal places. Solution EXAMPLE 8 (a) = ≈ log 89 log89 log5 2.7889 5 or = ≈ log 89 ln89 ln5 2.7889 5 (b) = ≈ log 5 log 5 log 2 2.3219 2 or = ≈ log 5 ln 5 ln 2 2.3219 2 Now Work PROBLEMS 23 AND 71 5 Graph a Logarithmic Function Whose Base Is Neither 10 Nor e The Change-of-Base Formula also can be used to graph logarithmic functions whose bases are neither 10 nor e .
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