485 4.4 Evaluating Logarithms and the Change-of-Base Theorem For each substance, find the pH from the given hydronium ion concentration to the nearest tenth. See Example 2(a). 29. grapefruit, 6.3 * 10-4 30. limes, 1.6 * 10-2 31. crackers, 3.9 * 10-9 32. sodium hydroxide (lye), 3.2 * 10-14 Find the 3H3O+4 for each substance with the given pH. Write answers in scientific notation to the nearest tenth. See Example 2(b). 33. soda pop, 2.7 34. wine, 3.4 35. beer, 4.8 36. drinking water, 6.5 Suppose that water from a wetland area is sampled and found to have the given hydronium ion concentration. Determine whether the wetland is a rich fen, a poor fen, or a bog. See Example 3. 37. 2.49 * 10-5 38. 6.22 * 10-5 39. 2.49 * 10-2 40. 3.14 * 10-2 41. 2.49 * 10-7 42. 5.86 * 10-7 Solve each problem. 43. Use a calculator to find an approximation for each logarithm. (a) log 398.4 (b) log 39.84 (c) log 3.984 (d) From the answers to parts (a)–(c), make a conjecture concerning the decimal values in the approximations of common logarithms of numbers greater than 1 that have the same digits. 44. Given that log 25 ≈1.3979, log 250 ≈2.3979, and log 2500 ≈3.3979, make a conjecture for an approximation of log 25,000. Why does this pattern continue? Find each value. If applicable, give an approximation to four decimal places. See Example 5. 45. ln e1.6 46. ln e 5.8 47. ln 1 e 2 48. ln 1 e 4 49. ln 2e 50. ln 23 e 51. ln 28 52. ln 39 53. ln 0.00013 54. ln 0.0077 55. ln 127 * 9432 56. ln 133 * 5682 57. ln 98 13 58. ln 84 17 59. ln 27 + ln 943 60. ln 33 + ln 568 61. ln 98 - ln 13 62. ln 84 - ln 17 Solve each problem. See Examples 4, 6, 7, and 9. 63. Decibel Levels Find the decibel ratings of sounds having the following intensities. (a) 100I0 (b) 1000I0 (c) 100,000I0 (d) 1,000,000I0 (e) If the intensity of a sound is doubled, by how much is the decibel rating increased? Round to the nearest whole number. 64. Decibel Levels Find the decibel ratings of the following sounds, having intensities as given. Round each answer to the nearest whole number. (a) whisper, 115I0 (b) busy street, 9,500,000I0 (c) heavy truck, 1,200,000,000I0 (d) rock music, 895,000,000,000I0 (e) jetliner at takeoff, 109,000,000,000,000I0
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