1 9 y f x x logx 1 and y g x x 464 exponential and

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1 (+) 0 0 ( ) 9 0 (+) y = f ( x ) = x log( x + 1) and y = g ( x ) = x
464 Exponential and Logarithmic Functions Our next example revisits the concept of pH as first introduced in the exercises in Section 6.1 . Example 6.4.3. In order to successfully breed Ippizuti fish the pH of a freshwater tank must be at least 7.8 but can be no more than 8.5. Determine the corresponding range of hydrogen ion concentration, and check your answer using a calculator. Solution. Recall from Exercise 77 in Section 6.1 that pH = log[H + ] where [H + ] is the hydrogen ion concentration in moles per liter. We require 7 . 8 ≤ − log[H + ] 8 . 5 or 7 . 8 log[H + ] ≥ − 8 . 5. To solve this compound inequality we solve 7 . 8 log[H + ] and log[H + ] ≥ − 8 . 5 and take the intersection of the solution sets. 3 The former inequality yields 0 < [H + ] 10 7 . 8 and the latter yields [H + ] 10 8 . 5 . Taking the intersection gives us our final answer 10 8 . 5 [H + ] 10 7 . 8 . (Your Chemistry professor may want the answer written as 3 . 16 × 10 9 [H + ] 1 . 58 × 10 8 .) After carefully adjusting the viewing window on the graphing calculator we see that the graph of f ( x ) = log( x ) lies between the lines y = 7 . 8 and y = 8 . 5 on the interval [3 . 16 × 10 9 , 1 . 58 × 10 8 ]. The graphs of y = f ( x ) = log( x ), y = 7 . 8 and y = 8 . 5 We close this section by finding an inverse of a one-to-one function which involves logarithms. Example 6.4.4. The function f ( x ) = log( x ) 1 log( x ) is one-to-one. Find a formula for f 1 ( x ) and check your answer graphically using your calculator. Solution. We first write y = f ( x ) then interchange the x and y and solve for y . y = f ( x ) y = log( x ) 1 log( x ) x = log( y ) 1 log( y ) Interchange x and y . x (1 log( y )) = log( y ) x x log( y ) = log( y ) x = x log( y ) + log( y ) x = ( x + 1) log( y ) x x + 1 = log( y ) y = 10 x x +1 Rewrite as an exponential equation. 3 Refer to page 4 for a discussion of what this means.
6.4 Logarithmic Equations and Inequalities 465 We have f 1 ( x ) = 10 x x +1 . Graphing f and f 1 on the same viewing window yields y = f ( x ) = log( x ) 1 log( x ) and y = g ( x ) = 10 x x +1
466 Exponential and Logarithmic Functions 6.4.1 Exercises In Exercises 1 - 24 , solve the equation analytically. 1. log(3 x 1) = log(4 x ) 2. log 2 ( x 3 ) = log 2 ( x ) 3. ln ( 8 x 2 ) = ln(2 x ) 4. log 5 ( 18 x 2 ) = log 5 (6 x ) 5. log 3 (7 2 x ) = 2 6. log 1 2 (2 x 1) = 3 7. ln ( x 2 99 ) = 0 8. log( x 2 3 x ) = 1 9. log 125 3 x 2 2 x + 3 = 1 3 10. log x 10 3 = 4 . 7 11. log( x ) = 5 . 4 12. 10 log x 10 12 = 150 13. 6 3 log 5 (2 x ) = 0 14. 3 ln( x ) 2 = 1 ln( x ) 15. log 3 ( x 4) + log 3 ( x + 4) = 2 16. log 5 (2 x + 1) + log 5 ( x + 2) = 1 17. log 169 (3 x + 7) log 169 (5 x 9) = 1 2 18. ln( x + 1) ln( x ) = 3 19. 2 log 7 ( x ) = log 7 (2) + log 7 ( x + 12) 20. log( x ) log(2) = log( x + 8) log( x + 2) 21. log 3 ( x ) = log 1 3 ( x ) + 8 22. ln(ln( x )) = 3 23. (log( x )) 2 = 2 log( x ) + 15 24. ln( x 2 ) = (ln( x )) 2 In Exercises 25 - 30 , solve the inequality analytically. 25. 1 ln( x ) x 2 < 0 26. x ln( x ) x > 0 27. 10 log x 10 12 90 28. 5 . 6 log x 10 3 7 . 1 29. 2 . 3 < log( x ) < 5 . 4 30. ln( x 2 ) (ln( x )) 2 In Exercises 31 - 34 , use your calculator to help you solve the equation or inequality.

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