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3. Splash Screen Topic 12

4. Contents Lesson 12-1Exponential Functions Lesson 12-2Logarithms and Logarithmic Functions Lesson 12-3Properties of Logarithms Lesson 12-4Common Logarithms Lesson 12-5Base e and Natural Logarithms Lesson 12-6Exponential Growth and Decay

5. Lesson 1 Contents Example 1Graph an Exponential Function Example 2Identify Exponential Growth and Decay Example 3Write an Exponential Function Example 4Simplify Expressions with Irrational Exponents Example 5Solve Exponential Equations Example 6Solve Exponential Inequalities

6. Sketch the graph of. Then state the function’s domain and range. Example 1-1a Make a table of values. Connect the points to sketch a smooth curve. 162 41 10 –1 –2 x

7. Example 1-1b Answer: The domain is all real numbers, while the range is all positive numbers.

8. Sketch the graph of Then state the function’s domain and range. Example 1-1c Answer: The domain is all real numbers; the range is all positive numbers.

9. Determine whether represents exponential growth or decay. Example 1-2a Answer: The function represents exponential decay, since the base, 0.7, is between 0 and 1. Using technology

10. Determine whether represents exponential growth or decay. Example 1-2b Answer: The function represents exponential growth, since the base, 3, is greater than 1.

11. Determine whether represents exponential growth or decay. Example 1-2c Answer: The function represents exponential growth, since the base, is greater than 1.

12. Determine whether each function represents exponential growth or decay. a. b. c. Example 1-2d Answer: The function represents exponential decay, since the base, 0.5, is between 0 and 1. Answer:The function represents exponential growth, since the base, 2, is greater than 1. Answer: The function represents exponential decay, since the base,is between 0 and 1.

13. Write an exponential function of the form that could be used to model the number of cellular telephone subscribers y in the U.S. Write the function in terms of x, the number of years since 1990. Example 1-3a Cellular Phones In December of 1990, there were 5,283,000 cellular telephone subscribers in the United States. By December of 2000, this number had risen to 109,478,000. For 1990, the time x equals 0, and the initial number of cellular telephone subscribers y is 5,283,000. Thus the y -intercept, and the value of a, is 5,283,000. For 2000, the time x equals 2000 – 1990 or 10, and the number of cellular telephone subscribers is 109,478,000.

14. Example 1-3b Substitute these values and the value of a into an exponential function to approximate the value of b. Exponential function Replace x with 10, y with 109,478,000 and a with 5,283,000. Divide each side by 5,283,000. Take the 10 th root of each side.

15. Example 1-3c ENTERMATH Keystrokes: 10520.72 1.354063324 To find the 10 th root of 20.72, use selection under the MATH menu on the TI-83/84 Plus. Answer: An equation that models the number of cellular telephone subscribers in the U.S. from 1990 to 2000 is

16. Example 1-3d Suppose the number of telephone subscribers continues to increase at the same rate. Estimate the number of US subscribers in 2010. For 2010, the time x equals 2010 – 1990 or 20. Answer: The number of cell phone subscribers will be about 2,136,000,000 in 2010. Modeling equation Replace x with 20. Use a calculator.

17. Health In 1991, 4.9% of Americans had diabetes. By 2000, this percent had risen to 7.3%. a.Write an exponential function of the form could be used to model the percentage of Americans with diabetes. Write the function in terms of x, the number of years since 1991. b.Suppose the percent of Americans with diabetes continues to increase at the same rate. Estimate the percent of Americans with diabetes in 2010. Example 1-3e Answer: 11.4 % Answer:

18. Example 1-4a Simplify. Quotient of PowersAnswer:

19. Example 1-4b Simplify.Power of a Power Product of RadicalsAnswer:

20. Example 1-4c Simplify each expression. a. b. Answer:

21. Example 1-5a Original equation Rewrite 256 as 4 4 so each side has the same base. Property of Equality for Exponential Functions Add 2 to each side. Divide each side by 9. Answer: The solution is Solve.

22. Example 1-5b Check Original equation Simplify. Substitute for n.

23. Example 1-5c Original equation Rewrite 9 as 3 2 so each side has the same base. Property of Equality for Exponential Functions Distributive Property Subtract 4x from each side. Answer: The solution is Solve.

24. Solve each equation. a. b. Example 1-5d Answer: 1 Answer:

25. Example 1-6a SolveOriginal inequality Property of Inequality for Exponential Functions Subtract 3 from each side. Divide each side by –2. Answer: The solution is Rewrite as

26. Example 1-6b Check: Test a value of k less than for example, Original inequality Replace k with 0. Simplify.

27. Example 1-6c Solve Answer:

28. End of Lesson 1 Topic 12 - 1

29. Lesson 2 Contents Example 1Logarithmic to Exponential Form Example 2Exponential to Logarithmic Form Example 3Evaluate Logarithmic Expressions Example 4Inverse Property of Exponents and Logarithms Example 5Solve a Logarithmic Equation Example 6Solve a Logarithmic InequalityExample 6Solve a Logarithmic Inequality (SKIP) Example 7Solve Equations with Logarithms on Each Side

30. Example 2-1a Write in exponential form. Answer:

31. Example 2-1b Write in exponential form. Answer:

32. Write each equation in exponential form. a. b. Example 2-1c Answer:

33. Example 2-2a Write in logarithmic form. Answer:

34. Example 2-2b Write in logarithmic form. Answer:

35. Example 2-2c Write each equation in logarithmic form. a. b. Answer:

36. Example 2-3a Evaluate Let the logarithm equal y. Definition of logarithm Property of Equality for Exponential Functions Answer: So,

37. Example 2-3b Answer: 3 Evaluate

38. Example 2-4a Evaluate. Answer:

39. Example 2-4b Evaluate. Answer:

40. Example 2-4c Evaluate each expression. a. b. Answer: 3 Answer:

41. Example 2-5a Solve Original equation Definition of logarithm Power of a Power Simplify. Answer:

42. Example 2-5b Solve Answer: 9

43. Example 2-6a SolveCheck your solution. Original inequality Logarithmic to exponential inequality Simplify. Answer: The solution set is

44. Example 2-6b Check Try 6 4 to see if it satisfies the inequality. Original inequality Substitute 6 4 for x.

45. Example 2-6c Solve Check your solution. Answer:

46. Example 2-7a Solve Check your solution. Original equation Property of Equality for Logarithmic Functions Subtract 4x and add 3 to each side. Factor. Solve each equation. Zero Product Property or

47. Example 2-7b Check Substitute each value into the original equation. Original equation Answer: The solutions are 3 and 1. Simplify. Substitute 3 for x. Substitute 1 for x.

48. Example 2-7c Answer: The solutions are 3 and –2. Solve Check your solution.

49. End of Lesson 2 Topic 12 - 2

50. Lesson 3 Contents Example 1Use the Product Property Example 2Use the Quotient Property Example 3Use Properties of Logarithms Example 4Power Property of Logarithms Example 5Solve Equations Using Properties of Logarithms

51. Example 3-1a Use to approximate the value of Replace with 0.4307. Answer: Thus, is approximately 3.4307. Product Property Inverse Property of Exponents and Logarithms Replace 250 with 5 3 2.

52. Example 3-1a Answer: Thus, is approximately 3.4307. First Property of Logarithms

53. Example 3-1b Answer: 6.5850 Use to approximate the value of

54. Example 3-2a Use andto approximate the value of Quotient Property and Answer: Thus is approximately 0.7737. Replace 4 with the quotient

55. Example 3-1a Second Property of Logarithms

56. Example 3-2b Answer: 1.2920 Use andto approximate the value of

57. Example 3-3a Sound The loudness L of a sound in decibels is given by where R is the sound’s relative intensity. The sound made by a lawnmower has a relative intensity of 10 9 or 90 decibels. Would the sound of ten lawnmowers running at that same intensity be ten times as loud or 900 decibels? Explain your reasoning. Let L 1 be the loudness of one lawnmower running. Let L 2 be the loudness of ten lawnmowers running.

58. Example 3-3b Then the increase in loudness is L 2 – L 1. Distributive Property Substitute for L 1 and L 2. Product Property Subtract. Inverse Property of Exponents and Logarithms

59. Example 3-3c Answer:No; the sound of ten lawnmowers is perceived to be only 10 decibels as loud as the sound of one lawnmower, or 100 decibels.

60. Example 3-3d Sound The loudness L of a sound in decibels is given by where R is the sound’s relative intensity. The sound made by fireworks has a relative intensity of 10 14 or 140 decibels. Would the sound of ten fireworks of that same intensity be ten times as loud or 1400 decibels? Explain your reasoning. Then the increase in loudness is L 2 – L 1.

62. Inverse Property of Exponents and Logarithms

63. Example 3-3d Sound The loudness L of a sound in decibels is given by where R is the sound’s relative intensity. The sound made by fireworks has a relative intensity of 10 14 or 140 decibels. Would the sound of ten fireworks of that same intensity be ten times as loud or 1400 decibels? Explain your reasoning. Answer:No; the sound of ten fireworks is perceived to be only 10 more decibels as loud as the sound of one firework, or 150 decibels.

64. Example 3-4a Given that approximate the value of Replace 216 with 6 3. Power Property Replace with 1.1133. Answer:

65. Example 3-1a Third Property of Logarithms: Floating Exponent

66. Example 3-4b Given that approximate the value of Answer: 5.1700

67. Example 3-5a Solve. Original equation Power Property Quotient Property Property of Equality for Logarithmic Functions Multiply each side by 5.

68. Example 3-5b Answer: Take the 4 th root of each side.

69. Example 3-5c Solve. Original equation Product Property Definition of logarithm Subtract 64 from each side. Factor. Solve each equation. Zero Product Propertyor

70. Example 3-5d Check Substitute each value into the original equation. Since log 8 (–4) and log 8 (–16) are undefined, –4 is an extraneous solution and must be eliminated. Replace x with –4. Replace x with 16. Product Property

71. Example 3-5d Answer: The only solution is Definition of logarithm

72. Solve each equation. a. b. Example 3-5e Answer: 12 Answer: 8

73. End of Lesson 3 Topic 12 - 3

74. Lesson 4 Contents Example 1Find Common Logarithms Example 2Solve Logarithmic Equations Using Exponentiation Example 3Solve Exponential Equations Using Logarithms Example 4Solve Exponential Inequalities Using Logarithms Example 5Change of Base Formula

75. Example 4-1a Use a calculator to evaluate log 6 to four decimal places. ENTERLOG Keystrokes: 6.7781512503 Answer: about 0.7782

76. Example 4-1b Use a calculator to evaluate log 0.35 to four decimal places. ENTERLOG Keystrokes: 0.35 –.4559319557 Answer: about –0.4559

77. Example 4-1c Use a calculator to evaluate each expression to four decimal places. a. log 5 b. log 0.62 Answer: 0.6990 Answer: –0.2076

78. Example 4-2a Earthquake The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log The San Fernando Valley earthquake of 1994 measured 6.6 on the Richter scale. How much energy did this earthquake release? Write the formula. Replace M with 6.6. Simplify. Write each side using 10 as a base.

79. Example 4-2b Answer: The amount of energy released was about ergs. Inverse Property of Exponents and Logarithms Use a calculator.

80. Example 4-2c Earthquake The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log In 1999 an earthquake in Turkey measured 7.4 on the Richter scale. How much energy did this earthquake release? Answer: about

81. Example 4-3a Solve Original equation Take the log 5 of both sides Property of Logarithms Change of base. Use a calculator. Answer:

82. Example 4-3b Check You can check this answer by using a calculator or by using estimation. Since and the value of x is between 2 and 3. Thus, 2.5643 is a reasonable solution.

83. Example 4-3c Answer: 2.5789 Solve

84. Example 4-4a Solve Original inequality Property of Inequality for Logarithmic Functions Power Property of Logarithms Distributive Property Subtract 5x log 3 from each side.

85. Example 4-4b Factor an x. Switch > to < because is negative. Divide each side by Use a calculator. Simplify.

86. Example 4-4d Check: Original inequality Answer: The solution set is Negative Exponent Property Replace x with 0. Simplify.

87. Example 4-4e Solve Original inequality Property of Inequality for Logarithmic Functions Power Property of Logarithms Distributive Property Subtract x log 10 from each side.

88. Example 4-4e Solve Subtract x log 10 from each side. Answer: Factor an x DON’T Switch from 3log5 – log10 IS NOT negative Divide both sides by (3log5 – log10)

89. Example 4-5a Express in terms of common logarithms. Then approximate its value to four decimal places. Answer: The value of is approximately 2.6309. Change of Base Formula Use a calculator.

90. Example 4-5b Express in terms of common logarithms. Then approximate its value to four decimal places. Answer:

91. End of Lesson 4 Topic 12 - 4

92. Lesson 5 Contents Example 1Evaluate Natural Base Expressions Example 2Evaluate Natural Logarithmic Expressions Example 3Write Equivalent Expressions Example 4Inverse Property of Base e and Natural Logarithms Example 5Solve Base e Equations Example 6Solve Base e Inequalities Example 7Solve Natural Log Equations and Inequalities

93. Example 5-1a Answer: about 1.6487 ENTER2nd Keystrokes: [e x ] 0.5 1.648721271 Use a calculator to evaluate to four decimal places.

94. Example 5-1b Answer: about 0.0003 ENTER2nd Keystrokes: Ti-83/84 e ^ ( –8).0003354626 Use a calculator to evaluate to four decimal places. LN

95. Example 5-1c Use a calculator to evaluate each expression to four decimal places. a. b. Answer: 1.3499 Answer: 0.1353

96. Example 5-2d Use a calculator to evaluate In 3 to four decimal places. Keystrokes: ENTERLN 3 1.098612289 Answer: about 1.0986

97. Example 5-2e Keystrokes: ENTERLN 1 ÷ 4 –1.386294361 Answer: about –1.3863 Use a calculator to evaluate In to four decimal places.

98. Use a calculator to evaluate each expression to four decimal places. a. In 2 b. In Example 5-2f Answer: 0.6931 Answer: –0.6931

99. Example 5-3a Answer: Write an equivalent logarithmic equation for.

100. Example 5-3b Answer: Write an equivalent exponential equation for

101. Example 5-3c Answer: Write an equivalent exponential or logarithmic equation. a. b.

102. Example 5-4a Evaluate, using your calculator. Answer:

103. Example 5-4b Evaluate. Answer:

104. Evaluate each expression. a. b. Example 5-4c Answer: 7 Answer:

105. Example 5-5a Solve Original equation Subtract 4 from each side. Divide each side by 3. Property of Equality for Logarithms Divide each side by –2. Use a calculator. Answer: The solution is about –0.3466. Inverse Property of Exponents and Logarithms

106. Example 5-5b Check You can check this value by substituting – 0.3466 into the original equation or by finding the intersection of the graphs of and

107. Example 5-5c Answer: 0.8047 Solve

108. Example 5-6a Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. What is the balance after 8 years? Answer: The balance after 8 years would be $1131.25. Continuous compounding formula Replace P with 700, r with 0.06, and t with 8. Simplify. Use a calculator.

109. Example 5-6b How long will it take for the balance in your account to reach at least $2000? Divide each side by 700. Property of Inequality for Logarithms The balance is at least $2000. Write an inequality. 2000 Replace A with 700e (0.06)t. A  Inverse Property of Exponents and Logarithms

110. Example 5-6c Answer: It will take at least 17.5 years for the balance to reach $2000. Use a calculator. Divide each side by 0.06.

111. Example 5-6d Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. a.What is the balance after 7 years? b.How long will it take for the balance in your account to reach at least $2500 ? Answer: $1065.37 Answer: at least 21.22 years

112. Example 5-7a Answer: The solution is 0.5496. Check this solution using substitution or graphing. Original equation Write each side using exponents and base e. Inverse Property of Exponents and Logarithms Use a calculator. Divide each side by 3. Solve

113. Example 5-7b Original inequality Write each side using exponents and base e. Inverse Property of Exponents and Logarithms Add 3 to each. Use a calculator. Divide each side by 2.

114. Example 5-7c Answer: The solution is all numbers less than 7.5912 and greater than 1.5. Check this solution using substitution.

115. Solve each equation or inequality. a. b. Example 5-7d Answer: about 1.0069 Answer:

116. End of Lesson 5 Topic 12 - 5

117. Lesson 6 Contents Example 1Exponential Decay of the Form y = a(1 – r) t Example 2Exponential Decay of the Form y = ae –kt Example 3Exponential Growth of the Form y = a(1 + r ) t Example 4Exponential Growth of the Form y = ae kt

118. Example 6-1a Caffeine A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for 90% of this caffeine to be eliminated from a person’s body? Explore The problem gives the amount of caffeine consumed and the rate at which the caffeine is eliminated. It asks you to find the time it will take for 90% of the caffeine to be eliminated from a person’s body. Use the formula Let t be the number of hours since drinking the coffee. The amount remaining y is 10% of 130 or 13. Plan

119. Example 6-1b Solve Exponential decay formula Replace y with 13, a with 130, and r with 0.11. Divide each side by 130. Property of Equality for Logarithms Power Property for Logarithms Divide each side by log 0.89. Use a calculator.

120. Example 6-1c Answer:It will take approximately 20 hours for 90% of the caffeine to be eliminated from a person’s body. Examine Use the formula to find how much of the original 130 milligrams of caffeine would remain after 20 hours. Ten percent of 130 is 13, so the answer seems reasonable. Exponential decay formula Replace a with 130, r with 0.11 and t with 20.

121. Example 6-1d Caffeine A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for 80% of this caffeine to be eliminated from a person’s body? Answer: 13.8 hours

122. Example 6-2a Geology The half-life of Sodium-22 is 2.6 years. What is the value of k for Sodium-22? Replace y with 0.5a and t with 2.6. Divide each side by a. Property of Equality for Logarithmic Functions Inverse Property of Exponents and Logarithms Divide each side by –2.6. Exponential decay formula

123. Example 6-2b Use a calculator. Answer: The constant k for Sodium-22 is 0.2666. Thus, the equation for the decay of Sodium-22 is where t is given in years.

124. Example 6-2c A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached the surface of the Earth. How long ago did the meteorite reach the surface of the Earth? Formula for the decay of Sodium- 22 Replace y with 0.1a. Divide each side by a. Property of Equality for Logarithms

125. Example 6-2d Answer: It was formed about 9 years ago. Inverse Property for Exponents and Logarithms Divide each side by –0.2666. Use a calculator.

126. Example 6-2e Health The half-life of radioactive iodine used in medical studies is 8 hours. a. What is the value of k for radioactive iodine? b.A doctor wants to know when the amount of radioactive iodine in a patient’s body is 20% of the original amount. When will this occur? Answer: about 19 hours later Answer:

127. You want to know when the population has doubled or is 2 million. Use the formula Example 6-3a Multiple-Choice Test Item The population of a city of one million is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled? A 4 yearsB 5 years C 20 yearsD 23 years Read the Test Item

128. Example 6-3b Solve the Test Item Exponential growth formula Replace y with 2,000,000, a with 1,000,000, and r with 0.03. Divide each side by 1,000,000. Property of Equality for Logarithms Power Property of Logarithms

129. Example 6-3c Answer: D Divide each side by ln 1.03. Use a calculator.

130. Example 6-3d Multiple-Choice Test Item The population of a city of 10,000 is increasing at a rate of 5% per year. If the population continues to grow at this rate, in how many years will the population have doubled? A 10 yearsB 12 years C 14 yearsD 18 years Answer: C

131. Example 6-4a You want to find t such that Population As of 2000, Nigeria had an estimated population of 127 million people and the United States had an estimated population of 278 million people. The growth of the populations of Nigeria and the United States can be modeled by and, respectively. According to these models, when will Nigeria’s population be more than the population of the United States? Replace N(t) with and U(t) with

132. Example 6-4b Answer:After 46 years or in 2046, Nigeria’s population will be greater than the population of the U.S. Inverse Property of Exponents and Logarithms Subtract ln 278 and 0.026t from each side. Divide each side by –0.017. Use a calculator. Product Property of Logarithms Property of Inequality for Logarithms

133. Example 6-4c Answer: after 109 years or in the year 2109 Population As of 2000, Saudi Arabia had an estimated population of 20.7 million people and the United States had an estimated population of 278 million people. The growth of the populations of Saudi Arabia and the United States can be modeled by and, respectively. According to these models, when will Saudi Arabia’s population be more than the population of the United States?

134. End of Lesson 6 Topic 12 - 6

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