MA460 Honors Precalculus Supplementary Problems: Exponential and Logarithmic Functions • Sketch the graph of each function. 1. f (x) = 2x!"#$% 1 2. ƒ(x) = log2 (9 – 2x) 3. ƒ(x) = 2x + 2x9 4. ƒ(x) = 9 – log2 (2x) • If necessary, restrict the domain of ƒ to make it a one-to-one function. Then find ƒ–1(x). 5. ƒ(x) = 2ln x + 12ln x ! 1 6. ƒ(x) = ex2! !1+ 1 7. ƒ(x) = ln (x2 – 16) 8. ƒ(x) = –1 + log3 (4x – 7) • Solve each equation for x: 9. 3x2= 7x log73 10. log5 x = logx 5 11. 32x – 3x+1 + 2 = 0 12. xlog x= 100x 13. log3 (x3 – 8) – log3 (x – 2) = 2 14. log4 25x16– 2 log4 5x2+ log4 32x5= 12 15. l og3(7x + 1) + log312x +67!"#$%&= 2 16. (log x)2 – log x = 2 17. log x2 – log x = 2 18. log3 121 – log3 x7= – 32 19. ex ln 2= 43 20. l og!(5x)3= 2 21. x = log3 729 91.274 / 334 22. x = log4 7516 – 2 log4 59 + log4 32243 23. 3x – 3–x = 31/2 (31/2 – 3–1/2) 24. logx 1 = 0 25. logx 5 = l og75l og73 26. 2 ! 52 x 3= 7 27. 2x + 2–x = 33 28 28. 3x+2 – 34–x = 728 • Solve each inequality for x: 29. 94!"#$%&x ' 3<23!"#$%&2 x '1 30. log2/5 (x + 2) > log2/5 3 31. log3 x < 2 32. log1/2 (x + 1) > log1/2 (3x + 2)33. Let f (x) =ex! e! x2, g(x) = ln (x + 1). (a) Find ƒ(0) and f l n75!"#$%&. (b) Find ƒ(g(x)) and simplify. (c) Find the domain of the function ƒ°g. 34. Let P0 be the initial population and P(t) the population at any time t ≥ 0 of some mammal in its niche. If environmental factors constrain the population growth in such a way that L is the largest population the niche can comfortably accommodate, then one model that sometimes describes the population growth or decay is P(t ) =L1 +L ! P0P0e! k L t. where k is a growth or decay factor. (a) If P0 = 100, L = 180, and P(3) = 160, find k. Then sketch the graph. (b) Let y = P(t). Find P!1(y). 35. Suppose ƒ(x ) = a . bx, ƒ–1(8/9) = 2, and ƒ–1(4/3) = 1. Find a and b. 36. Let ƒ(x) = 2x–3 + 1. Find log8 (ƒ(3x + 3) – 1). 37. Let ƒ(x) = ln (sec x + 1) + ln (sec x – 1), g(x) = 2 ln(tan x ). Find all x for which ƒ(x) = g(x). • Let r = log5 2, s = log5 3. In terms of r and s, find 38. log5 1.5 39. log5 12 40. log5 0.25 41. log5 183 42. log3 2 43. log2 5 • Let ƒ(x) = 10x and g(x) = log10 x. Which of the following equations are identities? 44. ƒ(a + b) = ƒ(a)ƒ(b) 45. ƒ(ab) = ƒ(a ) + ƒ(b) 46. g(a + b) = g(a)g(b) 47. g(ab) = g(a) + g(b) 48. g(ac) = c g(a), c ≠ 0 49. ƒ(g(a)) = a 50. [g(a)]c= g(ac), c ≠