SECTION 5.4 Logarithmic Functions 313 5.4 Logarithmic Functions Now Work the ‘Are You Prepared?’ problems on page 321. • Solve Linear Inequalities (Section A.9, pp. A80–A82) • Quadratic Inequalities (Section 3.5, pp. 180–181) • Polynomial and Rational Inequalities (Section 4.7, pp. 259–263) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements (p. 313) 2 Evaluate Logarithmic Expressions (p. 314) 3 Determine the Domain of a Logarithmic Function (p. 314) 4 Graph Logarithmic Functions (p. 315) 5 Solve Logarithmic Equations (p. 319) Recall that a one-to-one function ( ) = y f x has an inverse function that is defined implicitly by the equation ( ) = x f y . In particular, the exponential function ( ) = = y f x a ,x where > a 0 and ≠ a 1, is one-to-one, so it has an inverse function that is defined implicitly by the equation = > ≠ x a a a 0 1 y This inverse function is so important that it is given a name, the logarithmic function . DEFINITION Logarithmic Function with Base a The logarithmic function with base a , where > a 0 and ≠ a 1, is denoted by = y x loga (read as “ y is the logarithm with base a of x ”) and is defined by = = y x x a log if and only if a y The domain of the logarithmic function = y x loga is > x 0. As this definition illustrates, a logarithm is a name for a certain exponent . So x loga equals the exponent to which a must be raised to obtain x . 1 Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements The definition of a logarithm can be used to convert from exponential form to logarithmic form, and vice versa, as illustrated in the next two examples. Relating Logarithms to Exponents (a) If = y x log , 3 then = x 3 . y For example, the logarithmic statement = 4 log 81 3 is equivalent to the exponential statement = 81 3 .4 (b) If = y x log , 5 then = x 5 . y For example, ( ) − = 1 log 1 5 5 is equivalent to = − 1 5 5 .1 EXAMPLE 1 In Words When you need to evaluate x loga , think to yourself “ a raised to what power gives me x ?” Changing Exponential Statements to Logarithmic Statements Change each exponential statement to an equivalent statement involving a logarithm. (a) = m 1.23 (b) = e 9 b (c) = a 24 4 EXAMPLE 2 (continued)
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