832 Chapter 13 Exponential and Logarithmic Functions IMPORTANT FACTS AND CONCEPTS EXAMPLES Section 13.6 Properties for Solving Exponential and Logarithmic Equations a) If x = y, then ax = ay. a) If x = 5, then 3x = 35. b) If ax = ay, then x = y. b) If 3x = 35, then x = 5. c) If x = y, then logb x = logb y 1x 7 0, y 7 02. c) If x = 2, then log x = log 2. d) If logb x = logb y, then x = y 1x 7 0, y 7 02. d) If log x = log 2, then x = 2. Properties 6a-6d Section 13.7 Graph f1x2 = ex and g1x2 = ln x on the same set of axes. The natural exponential function is f1x2 = ex where e L 2.7183. y Natural logarithms are logarithms to the base e. Natural logarithms are indicated by the notation ln. 4 yx 3 loge x = ln x f (x) e x For x 7 0, if y = ln x, then e = x. y 2 1 4 3 2 1 1 2 The natural logarithmic function is 3 g1x2 = ln x 1 2 3 4 x g(x) ln x 4 where the base e L 2.7183. To approximate natural exponential and natural logarithmic values, use a scientific or graphing calculator. ln 5.83 L 1.7630 If ln x = -2.09, then x = e-2.09 = 0.1237. The natural exponential function, f1x2 = ex, and the natural logarithmic function, g1x2 = ln x, are inverses of each other. Change of Base Formula For any logarithm bases a and b, and positive number x, logb x loga x = logb a log5 98 = Properties for Natural Logarithms ln xy = ln x + ln y x ln = ln x - ln y y 1x 7 0 and y 7 02 Product rule 1x 7 0 and y 7 02 Quotient rule ln xn = n ln x 1x 7 02 Power rule log 98 L 2.8488 log 5 ln 7 # 30 = ln 7 + ln 30 ln x + 1 = ln 1x + 12 - ln 1x + 82 x + 8 ln m5 = 5 ln m Additional Properties for Natural Logarithms and Natural Exponential Expressions ln ex = x Property 7 ln e19 = 19 eln x = x, x 7 0 Property 8 eln 2 = 2 Chapter 13 Review Exercises [13.1] Given f1x2 = x2 - 3x + 4 and g1x2 = 2x - 5, find the following. 1. 1f g21x2 2. 1f g2132 3. 1g f21x2 4. 1g f21-32 Given f1x2 = 6x + 7 and g1x2 = 1x - 3, x 3, find the following. 5. 1f g21x2 6. 1g f21x2 05/05/2016 - RS0000000000000000000000176133 - Elementary & Intermediate Algebra for College Students, Media Update Chapter 13 Review Exercises 833 Determine whether each function is a one-to-one function. 7. 8. y 9. {16, 22, 14, 02, 1-5, 72, 13, 82} y x x 1 10. e 10, -22, 16, 12, 13, -22, a , 4 b f 2 12. y = x2 - 9 11. y = 1x + 8, x - 8 In Exercises 13 and 14, for each function, find the domain and range of both f1x2 and f-11x2. 13. {15, 32, 16, 22, 1-4, -32, 1-1, 82} 14. y 4 x In Exercises 15 and 16, find f-11x2 and graph f1x2 and f-11x2 on the same axes. 15. y = f1x2 = 4x - 2 16. y = f1x2 = 1 3x - 1 17. Yards to Feet The function f1x2 = 36x converts yards, x, into inches. Find the inverse function that converts inches into yards. In the inverse function, what do x and f-11x2 represent? 21. Compound Interest Jim Marino invested $1500 in a savings account that earns 4% interest compounded quarterly. Use r nt the compound interest formula A = pa1 + b to detern mine the amount in Jim's account after 5 years. 18. Gallons to Quarts The function f1x2 = 4x converts gallons, x, into quarts. Find the inverse function that converts quarts into gallons. In the inverse function, what do x and f-11x2 represent? [13.2] Graph the following functions. Thinkstock 19. y = 2x 1 x 20. y = a b 2 [13.3] Write each equation in logarithmic form. 22. 82 = 64 Write each equation in exponential form. 25. log2 32 = 5 24. 5-3 = 23. 811>4 = 3 26. log1>4 1 = 2 16 27. log6 1 125 1 = -2 36 Write each equation in exponential form and find the missing value. 28. 3 = log4 x 29. 4 = loga 81 30. -3 = log1>5 x Graph the following functions. 31. y = log3 x 32. y = log1>2 x 05/05/2016 - RS0000000000000000000000176133 - Elementary & Intermediate Algebra for College Students, Media Update 834 Chapter 13 Exponential and Logarithmic Functions [13.4] Use the properties of logarithms to expand each expression. 34. log3 1x - 9 33. log5 178 35. log 61a + 12 36. log 19 x4 712x + 325 Write the following as the logarithm of a single expression. 37. 5 log x - 3 log 1x + 12 39. 38. 41log 2 + log x2 - log y 1 [ln x - ln 1x + 22] - ln 2 3 40. 3 ln x + 1 ln 1x + 12 - 6 ln 1x + 42 2 Evaluate. 41. 8log 8 10 42. log4 45 43. 11log 9 81 44. log818 9 [13.5, 13.7] Use a calculator to approximate each logarithm. Round your answers to four decimal places. 45. log 819 46. ln 0.0281 Use a calculator to approximate each power of 10. Round your answers to four decimal places. 48. 103.157 47. 103.159 Use a calculator to find x. Round your answers to four decimal places. 49. log x = 4.063 50. log x = - 1.2262 Evaluate. 51. log 105 52. 10log 9 [13.6] Solve without using a calculator. 55. 625 = 5x 56. 49x = 1 7 53. 7 log 103.2 54. 2110log 4.72 57. 23x - 1 = 32 58. 27x = 32x + 5 61. 12.5x + 1 = 381 62. 3x + 2 = 8x Solve using a calculator. Round your answers to four decimal places. 59. 7x = 152 60. 3.1x = 856 Solve the logarithmic equation. 63. log7 12x - 32 = 2 64. log x + log 14x - 192 = log 5 65. log3 x + log3 12x + 12 = 1 66. ln 1x + 12 - ln 1x - 22 = ln 4 [13.7] Solve each exponential equation for the remaining variable. Round your answers to four decimal places. 68. 100 = A0 e-0.42132 67. 50 = 25e0.6t Solve for the indicated variable. 69. A = A0 ekt, for t 70. 200 = 800ekt, for k 71. ln y - ln x = 6, for y 72. ln 1y + 12 - ln 1x + 82 = ln 3, for y Use the change of base formula to evaluate. Write the answers rounded to four decimal places. 73. log2 196 74. log3 47 [13.2-13.7] 75. Compound Interest Find the amount of money accumulated if Justine Elwood puts $12,000 in a savings account yielding 6% interest compounded annually for 8 years. Use A = pa 1 + r nt b . n 76. Interest Compounded Continuously If $6000 is placed in a savings account paying 4% interest compounded continuously, find the time needed for the account to double in value. 77. Bacteria The bacteria Escherichia coli are commonly found in the bladders of humans. Suppose that 2000 bacteria are present at time 0. Then the number of bacteria present t minutes later may be found by the function N1t2 = 20001220.05t. a) When will 50,000 bacteria be present? b) Suppose that a human bladder infection is classified as a condition with 120,000 bacteria. When would a person develop a bladder infection if he or she started with 2000 bacteria? 05/05/2016 - RS0000000000000000000000176133 - Elementary & Intermediate Algebra for College Students, Media Update 835 Chapter 13 Practice Test 78. Atmospheric Pressure The atmospheric pressure, P, in pounds per square inch at an elevation of x feet above sea level can be found by the formula P = 14.7e-0.00004x. Find the atmospheric pressure at the top of Half Dome in Yosemite National Park, an elevation of 8842 feet. 79. Retention A class of history students is given a final exam at the end of the course. As part of a research project, the students are also given equivalent forms of the exam each month for n months. The average grade of the class after n months may be found by the function A1n2 = 72 - 18 log1n + 12, n 0. a) What was the class average when the students took the original exam 1n = 02? b) What was the class average for the exam given 3 months later? Allen R. Angel c) After how many months was the class average 58.0? Yosemite National Park Chapter 13 Practice Test Chapter Test Prep Videos provide fully worked-out solutions to any of the exercises you want to review. Chapter Test Prep Videos are available via , or on (search \"Angel Elementary and Intermediate Algebra\" and click on \"Channels\") 1. a) Determine whether the following function is a one-toone function. {14, 22, 1-3, 82, 1-1, 32, 16, -72} 14. Expand log2 x31x - 42 x + 2 . 15. Write as the logarithm of a single expression. b) List the set of ordered pairs in the inverse function. 2. Given f1x2 = x2 - 3 and g1x2 = x + 2, find a) 1f g21x2. b) 1f g2162. 3. Given f1x2 = x2 + 8 and g1x2 = 1x - 5, x 5, find a) 1g f21x2. b) 1g f2172. -1 -1 In Exercises 4 and 5, a) find f 1x2 and b) graph f1x2 and f 1x2 on the same axes. 4. y = f1x2 = -3x - 5 5. y = f1x2 = 1x - 1, x 1 6. What is the domain of y = log5 x? 7. Evaluate log4 1 . 256 8. Graph y = 3x. 9. Graph y = log2 x. 10. Write 2-5 1 = in logarithmic form. 32 11. Write log5 125 = 3 in exponential form. Write Exercises 12 and 13 in exponential form and find the missing value. 12. 4 = log2 1x + 32 13. y = log64 16 7 log6 1x - 42 + 2 log6 1x + 32 - 1 log6 x. 2 16. Evaluate 10 log9 19. 17. a) Find log 4620 rounded to 4 decimal places. b) Find ln 0.0692 rounded to 4 decimal places. 18. Solve 3x = 19 for x. 19. Solve log 4x = log 1x + 32 + log 2 for x. 20. Solve log 1x + 52 - log 1x - 22 = log 6 for x. 21. Find N, rounded to 4 decimal places, if ln N = 2.79. 22. Evaluate log6 40, rounded to 4 decimal places, using the change of base formula. 23. Solve 100 = 250e-0.03t for t, rounded to 4 decimal places. 24. Savings Account What amount of money accumulates if Kim Lee puts $3500 in a savings account yielding 4% interest compounded quarterly for 10 years? 25. Carbon 14 The amount of carbon 14 remaining after t years is found by the formula v = v0 e-0.0001205t, where v0 is the original amount of carbon 14. If a fossil originally contained 60 grams of carbon 14, and now contains 40 grams of carbon 14, how old is the fossil? 05/05/2016 - RS0000000000000000000000176133 - Elementary & Intermediate Algebra for College Students, Media Update