336 CHAPTER 5 Exponential and Logarithmic Functions Solving a Logarithmic Equation Solve: = x 2 log log 9 5 5 EXAMPLE 1 OBJECTIVES 1 Solve Logarithmic Equations (p. 336) 2 Solve Exponential Equations (p. 338) 3 Solve Logarithmic and Exponential Equations Using a Graphing Utility (p. 341) 5.6 Logarithmic and Exponential Equations Now Work the ‘Are You Prepared?’ problems on page 341. • Solve Equations Using a Graphing Utility (Section 1.4, pp. 28–31) • Solve Quadratic Equations (Section A.6, pp.A47–A53) • Solve Equations Quadratic in Form (Section A.6, pp. A53–A54) PREPARING FOR THIS SECTION Before getting started, review the following: 1 Solve Logarithmic Equations In Section 5.4 we solved logarithmic equations by changing a logarithmic equation to an exponential equation. That is, we used the definition of a logarithm: = = > ≠ y x x a a a log is equivalent to 0, 1 a y For example, to solve the equation ( ) − = x log 1 2 3, 2 use the equivalent exponential equation − = x 1 2 23 and solve for x . ( ) − = − = − = =− x x x x log 1 2 3 1 2 2 2 7 7 2 2 3 You should check this solution for yourself. For most logarithmic equations, some manipulation of the equation (usually using properties of logarithms) is required to obtain a solution. Also, to avoid extraneous solutions with logarithmic equations, determine the domain of the variable first. Our practice will be to solve equations, whenever possible, by finding exact solutions using algebraic methods and exact or approximate solutions using a graphing utility. When algebraic methods cannot be used, approximate solutions will be obtained using a graphing utility. You should pay particular attention to the form of equations for which exact solutions are possible. Let’s begin with an example of a logarithmic equation that requires using the fact that a logarithmic function is a one-to-one function. Simplify. Divide both sides by 2. − Change to exponential form. = = ≠ M N M N M N a a If log log , then , , and are positive and 1 a a
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