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Advanced Functions 4.3 NAME ____________________ DATE __________ PER _____ Logarithmic Functions and Their Applications Remember: g ( x) b x is a one-to-one function and has an inverse function. EX 1] Find the inverse of g ( x) b x . Definition of a Logarithm and a Logarithmic Function If x 0 and b is a positive constant b 1 , then y log b x if and only if The notation log b x is read “the logarithm (or log) base b of x”. The function defined by f ( x) log b x is a logarithmic function with base b. This function is the inverse of g ( x) b x . f ( x) log b x replaces the phrase “the power of b that produces x”. A logarithm is an exponent! Exponential Form and Logarithmic Form: The exponential form of y log b x is b y x . The logarithmic form of b y x is y log b x . EX 2] Write each equation in its exponential form. Logarithms are EXPONENTS! a) 3 log 2 8 b) 2 log 10 ( x 5) c) log e x 4 d) log b b3 3 EX 3] Write each equation in its logarithmic form. a) 32 9 b) 53 x c) a b c d) blogb 5 5 by x. Logarithmic Properties: 1) log b b 1 2) log b 1 0 3) log b b x x 4) b logb x x EX 4] Evaluate each of the following logarithms. a) log 8 1 b) log 5 5 c) log 2 24 d) 3log3 7 e) log 5 625 f) log 2 5 625 16 EX 5] Graph the logarithmic function f ( x) log 2 x . HINT: Do Loop-d-Loop first. Then, pick values for y and figure out x. x 2y x y x, y 2 1 x 0 1 2 EX 6] Graph the logarithmic function h( x) log 2 x . 3 HINT: Rewrite the function in its exponential form. y x Summary of the Properties of f ( x) log b x For positive real numbers b, b 1 , the function f ( x) log b x has the following properties: 1) The function f is a one-to-one function. Domain: __________________________ Range: __________________________ 2) The graph of f is a smooth continuous curve with a x-intercept of __________ , and the graph passes through __________. 3) If b 1 , f is an increasing function and the graph of f is asymptotic to the negative y-axis. This means As x , f ( x) _____ as x 0 from the RIGHT , , and f ( x) _____ . 4) If 0 b 1 , f is a decreasing function and the graph of f is asymptotic to the positive y-axis. This means As x , f ( x) _____ , and as x 0 from the RIGHT , f ( x) _____ . y y f ( x) log b x x f ( x) log b x x 0 b 1 b 1 EX 7] What is the domain of the graph of each of the following? a) f ( x) log 6 ( x 3) b) g ( x) ln x 4 x 2 x c) h( x) log 5 8 x EX 8] Explain how to use the graph of f ( x) log 4 x to produce the graph of g ( x) log 4 ( x 3) . y x EX 9] Explain how to use the graph of f ( x) log 4 x to produce the graph of h( x) log 4 x 3 . y x Definition of Common Logarithms: logarithms of base 10 EX] f ( x) log 10 x log x Definition of Natural Logarithms: logarithms of base e EX] f ( x) log e x ln x