SECTION 5.6 Logarithmic and Exponential Equations 337 Now Work PROBLEM 17 Graphing Solution To solve the equation using a graphing utility, graph = = Y x x 2 log 2 log log5 1 5 and = = Y log 9 log9 log5 , 2 5 and determine the point of intersection. See Figure 46 using a TI-84 Plus CE. The point of intersection is ( ) 3, 1.3652124 ; so = x 3 is the only solution. The solution set is { }3 . CAUTION A negative solution is not automatically extraneous. You must determine whether the potential solution causes the argument of any logarithmic expression in the equation to be negative or 0. j Now Work PROBLEM 25 Solving a Logarithmic Equation Solve: ( ) ( ) + + + = x x log 6 log 2 1 5 5 Algebraic Solution The domain of the variable requires that + > x 6 0 and + > x 2 0, so > − x 6 and > − x 2. This means any solution must satisfy > − x 2. To obtain an exact solution, first express the left side as a single logarithm. Then change the equation to exponential form. ( ) ( ) ( )( ) [ ] ( )( ) ( )( ) + + + = + + = + + = = + + = + + = + + = = − = − x x x x x x x x x x x x x x log 6 log 2 1 log 6 2 1 6 2 5 5 8 12 5 8 7 0 7 1 0 7 or 1 5 5 5 1 2 2 Only = − x 1 satisfies the restriction that > − x 2, so = − x 7 is extraneous. The solution set is { } −1 , which you should check. Graphing Solution Graph ( ) ( ) ( ) = + + + = f x x x log 6 log 2 5 5 ( ) ( ) + + + x x log 6 log 5 log 2 log 5 and ( ) = g x 1 using Desmos, and determine the point(s) of intersection. See Figure 47. The point of intersection is ( ) −1, 1 , so = − x 1 is the only solution. The solution set is { } −1 . Algebraic Solution The domain of the variable in this equation is > x 0. Note that each logarithm is to the same base, 5. Find the exact solution as follows: = = = = = − x x x x x 2 log log 9 log log 9 9 3 or 3 5 5 5 2 5 2 Check: 2log 3 log 9 log 3 log 9 log 9 log 9 5 ? 5 5 2 ? 5 5 5 = = = = r M M log log a a r The solution set is { }3 . = M r M log log a r a If =M N log log a a , then M N. = The domain of the variable is x 0. > Therefore, −3 is extraneous and must be discarded. Figure 46 3 22 22 6 Y2 5 log5 9 Y1 5 2 log5 x EXAMPLE 2 Factor. M N MN log log log a a a( ) + = Change to exponential form. Multiply out. Place the quadratic equation in standard form. Use the Zero-Product Property. Figure 47
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