1. Slide 3- 1

2. Chapter 3 Exponential, Logistic, and Logarithmic Functions

3. 3.1 Exponential and Logistic Functions

4. Slide 3- 4 Quick Review

5. Slide 3- 5 Quick Review Solutions

6. Slide 3- 6 What you’ll learn about Exponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models … and why Exponential and logistic functions model many growth patterns, including the growth of human and animal populations.

7. Slide 3- 7 Exponential Functions

8. Slide 3- 8 Exponential Functions Rules For Exponents If a > 0 and b > 0, the following hold true for all real numbers x and y.

9. Slide 3- 9 Use the rules for exponents to solve for x 4 x = 128 (2) 2x = 2 7 2x = 7 x = 7/2 2 x = 1/32 2 x = 2 -5 x = -5 Exponential Functions

10. Slide 3- 10 (x 3 y 2/3 ) 1/2 x 3/2 y 1/3 27 x = 9 -x+1 (3 3 ) x = (3 2 ) -x+1 3 3x = 3 -2x+2 3x = -2x+ 2 5x = 2 x = 2/5 Exponential Functions

11. Slide 3- 11 Example Finding an Exponential Function from its Table of Values

12. Slide 3- 12 Example Finding an Exponential Function from its Table of Values

13. Slide 3- 13 5432154321 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x y = 2 x If b > 1, then the graph of b x will: Rise from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) Exponential Functions

14. Slide 3- 14 5432154321 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x y = ( 1 / 2 ) x If 0 < b < 1, then the graph of b x will: Fall from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) Exponential Functions

15. Slide 3- 15 Example Transforming Exponential Functions Describe how to transform the graph of f(x) = 2 x into the graph g(x) = 2 x-2 The graph of g(x) = 2 x-2 is obtained by translat ing the graph of f(x) = 2 x by 2 units to the right.

16. Slide 3- 16 Example Transforming Exponential Functions

17. Slide 3- 17 Example Transforming Exponential Functions

18. Slide 3- 18 The Natural Base e

19. Slide 3- 19 Exponential Functions and the Base e

20. Slide 3- 20 Exponential Functions and the Base e

21. Slide 3- 21 Example Transforming Exponential Functions

22. Slide 3- 22 Example Transforming Exponential Functions

23. Slide 3- 23 Logistic Growth Functions

24. Slide 3- 24 Exponential Growth and Decay

25. Slide 3- 25 Exponential Functions Definitions Exponential Growth and Decay The function y = k a x, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. y new amount y O original amount b base t time h half life

26. Slide 3- 26 Exponential Functions An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g. (a)Find the amount remaining after t hours. (b)Find the amount remaining after 60 hours. a. y = y o b t/h y = 2 (1/2) (t/15) b. y = y o b t/h y = 2 (1/2) (60/15) y = 2(1/2) 4 y =.125 g

27. Slide 3- 27 Exponential Functions A bacteria double every three days. There are 50 bacteria initially present (a)Find the amount after 2 weeks. (b)When will there be 3000 bacteria? a. y = y o b t/h y = 50 (2) (14/3) y = 1269 bacteria

28. Slide 3- 28 Exponential Functions A bacteria double every three days. There are 50 bacteria initially present When will there be 3000 bacteria? b. y = y o b t/h 3000 = 50 (2) (t/3) 60 = 2 t/3

29. 3.2 Exponential and Logistic Modeling

30. Slide 3- 30 Quick Review

31. Slide 3- 31 Quick Review Solutions

32. Slide 3- 32 What you’ll learn about Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models … and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.

33. Slide 3- 33 Constant Percentage Rate Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown.

34. Slide 3- 34 Exponential Population Model

35. Slide 3- 35 Example Finding Growth and Decay Rates

36. Slide 3- 36 Example Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.

37. Slide 3- 37 Example Modeling Bacteria Growth

38. Slide 3- 38 Example Modeling Bacteria Growth

39. Slide 3- 39 Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

40. Slide 3- 40 Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003.

41. Slide 3- 41 Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.

42. Slide 3- 42 Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor which spreads logistically so that s(t) = 1500/(1 + 29 e.-.09t ) models the number of students who have heard the rumor at the end of t days, where t = 0 is the day the rumor begins to spread (a)How many students have heard the rumor by the end of Day 0? (b)How long does it take for 1000 students to hear the rumor?

43. Slide 3- 43 Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor which spreads logistically so that s(t) = 1500/(1 + 29 e.-.09t ) models the number of students who have heard the rumor at the end of t days, where t = 0 is the day the rumor begins to spread (a)How many students have heard the rumor by the end of Day 0? (b)How long does it take for 1000 students to hear the rumor?

44. 3.3 Logarithmic Functions and Their Graphs

45. Slide 3- 45 Quick Review

46. Slide 3- 46 Quick Review Solutions

47. Slide 3- 47 What you’ll learn about Inverses of Exponential Functions Common Logarithms – Base 10 Natural Logarithms – Base e Graphs of Logarithmic Functions Measuring Sound Using Decibels … and why Logarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.

48. Slide 3- 48 Logarithmic Functions The inverse of an exponential function is called a logarithmic function. Definition: x = a y if and only if y = log a x

49. Slide 3- 49 Changing Between Logarithmic and Exponential Form

50. Slide 3- 50 Logarithmic Functions log 4 16 = 2 ↔ 4 2 = 16 log 3 81 = 4 ↔ 3 4 = 81 log 10 100 = 2 ↔ 10 2 = 100

51. Slide 3- 51 Inverses of Exponential Functions

52. Slide 3- 52 Logarithmic Functions The function f (x) = log a x is called a logarithmic function. Domain: (0, ∞) Range: (-∞, ∞) Asymptote: x = 0 Increasing for a > 1 Decreasing for 0 < a < 1 Common Point: (1, 0)

53. Slide 3- 53 Basic Properties of Logarithms

54. Slide 3- 54 An Exponential Function and Its Inverse

55. Slide 3- 55 Common Logarithm – Base 10 Logarithms with base 10 are called common logarithms. The common logarithm log 10 x = log x. The common logarithm is the inverse of the exponential function y = 10 x.

56. Slide 3- 56 Basic Properties of Common Logarithms

57. Slide 3- 57 Example Solving Simple Logarithmic Equations

58. Slide 3- 58 Basic Properties of Natural Logarithms

59. Slide 3- 59 Graphs of the Common and Natural Logarithm

60. Slide 3- 60 Example Transforming Logarithmic Graphs

61. Slide 3- 61 Example Transforming Logarithmic Graphs

62. Slide 3- 62 Decibels

63. 3.4 Properties of Logarithmic Functions

64. Slide 3- 64 Quick Review

65. Slide 3- 65 Quick Review Solutions

66. Slide 3- 66 What you’ll learn about Properties of Logarithms Change of Base Graphs of Logarithmic Functions with Base b Re-expressing Data … and why The applications of logarithms are based on their many special properties, so learn them well.

67. Slide 3- 67 1. log a (a x ) = x for all x   2. a log a x = x for all x > 0 3. log a (xy) = log a x + log a y 4. log a (x/y) = log a x – log a y 5. log a x n = n log a x Common Logarithm: log 10 x = log x Natural Logarithm: log e x = ln x All the above properties hold. Logarithmic Functions

68. Slide 3- 68 Properties of Logarithms

69. Slide 3- 69 Example Proving the Product Rule for Logarithms

70. Slide 3- 70 Example Proving the Product Rule for Logarithms

71. Slide 3- 71 Example Expanding the Logarithm of a Product

72. Slide 3- 72 Example Expanding the Logarithm of a Product

73. Slide 3- 73 Example Condensing a Logarithmic Expression

74. Slide 3- 74 Example Condensing a Logarithmic Expression

75. Slide 3- 75 Change-of-Base Formula for Logarithms

76. Slide 3- 76 Example Evaluating Logarithms by Changing the Base

77. Slide 3- 77 Example Evaluating Logarithms by Changing the Base

78. 3.5 Equation Solving and Modeling

79. Slide 3- 79 Quick Review

80. Slide 3- 80 Quick Review Solutions

81. Slide 3- 81 What you’ll learn about Solving Exponential Equations Solving Logarithmic Equations Orders of Magnitude and Logarithmic Models Newton’s Law of Cooling Logarithmic Re-expression … and why The Richter scale, pH, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions.

82. Slide 3- 82 One-to-One Properties

83. Slide 3- 83 Example Solving an Exponential Equation Algebraically

84. Slide 3- 84 Example Solving an Exponential Equation Algebraically

85. Slide 3- 85 Example Solving a Logarithmic Equation

86. Slide 3- 86 Example Solving a Logarithmic Equation

87. Slide 3- 87 Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. Orders of magnitude can be used to compare any like quantities: A kilometer is 3 orders of magnitude longer than a meter. A dollar is 2 orders of magnitude greater than a penny. New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8.

88. Slide 3- 88 Richter Scale

89. Slide 3- 89 5.5 Graphs of Logarithmic Functions What is the magnitude on the Richter scale of an earthquake if a = 300, T = 30 and B = 1.2?

90. Slide 3- 90 pH In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H + ]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration: pH=-log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

91. Slide 3- 91 Newton’s Law of Cooling

92. Slide 3- 92 Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

93. Slide 3- 93 Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

94. Slide 3- 94 Regression Models Related by Logarithmic Re-Expression Linear regression:y = ax + b Natural logarithmic regression:y = a + blnx Exponential regression:y = a·b x Power regression:y = a·x b

95. Slide 3- 95 Three Types of Logarithmic Re-Expression

96. Slide 3- 96 Three Types of Logarithmic Re-Expression (cont’d)

97. Slide 3- 97 Three Types of Logarithmic Re-Expression (cont’d)

98. 3.6 Mathematics of Finance

99. Slide 3- 99 Quick Review

100. Slide 3- 100 Quick Review Solutions

101. Slide 3- 101 What you’ll learn about Interest Compounded Annually Interest Compounded k Times per Year Interest Compounded Continuously Annual Percentage Yield Annuities – Future Value Loans and Mortgages – Present Value … and why The mathematics of finance is the science of letting your money work for you – valuable information indeed!

102. Slide 3- 102 Interest Compounded Annually

103. Slide 3- 103 Interest Compounded k Times per Year

104. Slide 3- 104 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

105. Slide 3- 105 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years.

106. Slide 3- 106 Compound Interest – Value of an Investment

107. Slide 3- 107 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

108. Slide 3- 108 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.

109. Slide 3- 109 Annual Percentage Yield A common basis for comparing investments is the annual percentage yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period.

110. Slide 3- 110 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

111. Slide 3- 111 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?

112. Slide 3- 112 Future Value of an Annuity

113. Slide 3- 113 Present Value of an Annuity

114. Slide 3- 114 Chapter Test

115. Slide 3- 115 Chapter Test

116. Slide 3- 116 Chapter Test

117. Slide 3- 117 Chapter Test Solutions

118. Slide 3- 118 Chapter Test Solutions

119. Slide 3- 119 Chapter Test Solutions