SECTION 5.6 Logarithmic and Exponential Equations 341 3 Solve Logarithmic and Exponential Equations Using a Graphing Utility The algebraic techniques introduced in this section to obtain exact solutions apply only to certain types of logarithmic and exponential equations. Solutions for other types are usually studied in calculus, using numerical methods. For such types, we can use a graphing utility to approximate the solution. 1. Solve − − = x x7 30 0. 2 (pp. A47–A53) 2. Solve ( ) ( ) + − + + = x x 3 4 3 3 0. 2 (pp. A53–A54) 3. Approximate the solution(s) to = − x x 5 3 2 using a graphing utility. (pp. 28–31) 4. Approximate the solution(s) to − + = x x2 2 0 3 using a graphing utility. (pp. 28–31) Solving Equations Using a Graphing Utility Solve: + = x e 2 x Express the solution(s) rounded to two decimal places. Solution The solution is found using a TI-84 Plus CE by graphing = + Y x ex 1 and = Y 2 2 as shown in Figure 53(a). Note that because Y1 is an increasing function (do you know why?), there is only one point of intersection for Y1 and Y .2 Using the INTERSECT command reveals that the solution is 0.44, rounded to two decimal places. Figure 53(b) shows the solution using Desmos. Now Work PROBLEM 75 EXAMPLE 8 Figure 53 (a) TI-84 Plus CE 4 0 1 Y1 5 x 1 e x Y2 5 2 (b) Desmos 5.6 Assess Your Understanding ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. Skill Building In Problems 5–44, solve each logarithmic equation. Express irrational solutions in exact form. 5. = x log 2 4 6. ( ) + = x log 6 1 7. ( ) = x log 5 4 2 8. ( ) − = x log 3 1 2 3 9. ( ) + = x log 12 log 7 4 4 10. ( ) + = x log 2 11 log 3 5 5 11. = x log 3 4 12. − = x log 7 4 2 13. − = x log 2 1 log 13 5 5 14. + = − x x log 3 4 log5 12 9 9 15. = x 1 2 log 3log 2 7 7 16. − = x 2 log log 9 4 4 17. = − x 3log log 27 2 2 18. = x 2 log 3log 4 5 5 19. ( ) + + = x 2 log 5 log 9 2 6 6 20. ( ) + − = x 2 log 20 log 25 2 3 3 21. ( ) + + = x x log log 15 2 22. ( ) + − = x x log log 21 2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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