In Problems 14–19, solve each equation. Express irrational solutions in exact form. 14. = + 5 125 x 2 15. ( ) + = x log 9 2 16. − = −e 8 2 4 x 17. ( ) ( ) + = + x x log 3 log 6 2 18. = + e 7x x 3 19. ( ) ( ) − + + = x x log 4 log 4 3 2 2 20. Write ( ) − − x x x log 4 3 18 2 3 2 as the sum and/or difference of logarithms. Express powers as factors. 21. A 50-mg sample of a radioactive substance decays to 34 mg after 30 days. How long will it take for there to be 2 mg remaining? 22. (a) If $1000 is invested at 5% compounded monthly, how much is there after 8 months? (b) If you want to have $1000 in 9 months, how much do you need to place in a savings account now that pays 5% compounded quarterly? (c) How long does it take to double your money if you can invest it at 6% compounded annually? 23. The decibel level, D, of sound is given by the equation = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ D I I 10 log 0 where I is the intensity of the sound and = − I 10 0 12 watt per square meter. (a) If the shout of a single person measures 80 decibels, how loud would the sound be if two people shout at the same time? That is, how loud would the sound be if the intensity doubled? (b) The pain threshold for sound is 125 decibels. If the Athens Olympic Stadium 2004 (Olympiako Stadio Athinas ‘Spyros Louis’) can seat 74,400 people, how many people in the crowd need to shout at the same time for the resulting sound level to meet or exceed the pain threshold? (Ignore any possible sound dampening.) 1. Is the following graph the graph of a function? If it is, is the function one-to-one? x y 4 –4 4 –4 2. For the function ( ) = − + f x x x 2 3 1, 2 find: (a) ( ) f 3 (b) ( ) − f x (c) ( ) + f x h 3. Determine which points are on the graph of + = x y 1. 2 2 (a) ( ) 1 2 , 1 2 (b) ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 1 2 , 3 2 4. Solve the equation ( ) ( ) − = + x x 3 2 4 5 . 5. Graph the line − = x y 2 4 16. 6. (a) Graph the quadratic function ( ) = − + − f x x x2 3 2 by determining whether its graph is concave up or concave down and by finding its vertex, axis of symmetry, yintercept, and x-intercept(s), if any. (b) Solve ( ) ≤ f x 0. 7. Determine the quadratic function whose graph is given in the figure. y x 50 –10 (0, 24) Vertex: (4, –8) 8 4 –2 8. Graph ( ) ( ) = + − f x x 3 1 2 3 using transformations. 9. Given that ( ) = + f x x 2 2 and ( ) = − g x x 2 3 , find ( )( ) f g x and state its domain. What is ( )( ) f g 5 ? 10. For the polynomial function ( ) = − − + f x x x x x 3 15 12 60 4 3 2 (a) Determine the end behavior of the graph. (b) Find the x- and y-intercepts of the graph. (c) Find the real zeros and their multiplicity, and determine if the graph crosses or touches the x-axis at each intercept. (d) Determine the maximum number of turning points on the graph. (e) Graph the function. 11. For the function ( ) = + g x 3 2: x (a) Graph g using transformations. State the domain, range, and horizontal asymptote of the graph of g. (b) Determine the inverse of g. State the domain, range, and vertical asymptote of the graph of −g .1 (c) On the same coordinate axes as g, graph −g .1 12. Solve the equation: = − 4 8 x x 3 2 13. Solve the equation: x x log 1 log 2 3 log9 3 3 9 ( ) ( ) + + − = 14. Suppose that ( ) ( ) = + f x x log 2 . 3 Solve: (a) ( ) = f x 0 (b) ( ) > f x 0 (c) ( ) = f x 3 15. Data Analysis The following data represent the percent of all drivers by age who have been stopped by the police for any reason within the past year. The median age represents the midpoint of the upper and lower limit for the age range. Age Range Median Age, x Percent Stopped, y 16–19 17.5 18.2 20–29 24.5 16.8 30–39 34.5 11.3 40–49 44.5 9.4 50–59 54.5 7.7 ≥ 60 69.5 3.8 Cumulative Review 380 CHAPTER 5 Exponential and Logarithmic Functions
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