314 CHAPTER 5 Exponential and Logarithmic Functions Solution Use the fact that = y x loga and = x a , y where > a 0 and ≠ a 1, are equivalent. (a) If = m 1.2 , 3 then = m 3 log . 1.2 (b) If = e 9, b then = b log 9. e (c) If = a 24, 4 then = 4 log 24. a Now Work PROBLEM 13 Changing Logarithmic Statements to Exponential Statements Change each logarithmic statement to an equivalent statement involving an exponent. (a) = log 4 5 a (b) = − b log 3 e (c) = c log 53 Solution EXAMPLE 3 (a) If = log 4 5, a then = a 4. 5 (b) If = − b log 3, e then = −e b. 3 (c) If = c log 5 , 3 then = 3 5. c Now Work PROBLEM 21 2 Evaluate Logarithmic Expressions To find the exact value of a logarithm, write the logarithm in exponential notation using the fact that = y x loga is equivalent to = a x, y and use the fact that if = a a , u v then = u v. Now Work PROBLEM 29 3 Determine the Domain of a Logarithmic Function The logarithmic function = y x loga has been defined as the inverse of the exponential function = y a .x That is, if ( ) = f x a ,x then ( ) = −f x x log . a 1 Based on the discussion in Section 5.2 on inverse functions, for a function f and its inverse −f ,1 = = − − f f f f Domain of Range of and Range of Domain of 1 1 Consequently, it follows that • ( ) = = ∞ Domain of the logarithmic function Range of the exponential function 0, • ( ) = = −∞ ∞ Range of the logarithmic function Domain of the exponential function , The next box summarizes some properties of the logarithmic function. = = < <∞ −∞< <∞ y x x a x y log if and only if Domain: 0 Range: a y Finding the Exact Value of a Logarithmic Expression Find the exact value of: (a) log 16 2 (b) log 1 27 3 Solution EXAMPLE 4 (a) To find log 16, 2 think “2 raised to what power equals 16?” Then, = = = = y y log 16 2 16 2 2 4 y y 2 4 Therefore, = log 16 4. 2 (b) To find log 1 27 3 , think “3 raised to what power equals 1 27 ?” Then, = = = = − − y y log 1 27 3 1 27 3 3 3 y y 3 3 Therefore, = − log 1 27 3. 3 Change to exponential form. = 16 24 Equate exponents. Change to exponential form. = = − 1 27 1 3 3 3 3 Equate exponents.
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