1. Chapter 3 Exponential, Logistic, and Logarithmic Functions

2. Quick Review

3. Quick Review Solutions

4. Exponential Functions

5. Determine if they are exponential functions
𝑓 𝑥 = 4 𝑥
𝑔 𝑥 =6 𝑥 −9
𝑡 𝑥 =−2∗ 1.5 𝑥
ℎ 𝑥 =7∗ 3 −𝑥
𝑞 𝑥 =5∗ 6 3

6. Answers
Yes No no

7. Sketch an exponential function

8. Example Finding an Exponential Function from its Table of Values

9. Example Finding an Exponential Function from its Table of Values

10. Exponential Growth and Decay

11. Sketch exponential graph and determine if they are growth or decay
𝑓 𝑥 = 2 𝑥
𝑔 𝑥 = 𝑥
ℎ 𝑥 = 4 −𝑥

12. Example Transforming Exponential Functions

13. Example Transforming Exponential Functions

14. Example Transforming Exponential Functions

15. Group Activity Use this formula 1+ 1 𝑥 𝑥 Group 1 calculate when x=1
What do you guys notice?

16. The Natural Base e

17. Exponential Functions and the Base e

18. Exponential Functions and the Base e

19. Example Transforming Exponential Functions

20. Example Transforming Exponential Functions

21. Logistic Growth Functions

22. Example: Graph and Determine the horizontal asymptotes
𝑓 𝑥 = 7 1+3∗ .6 𝑥

23. Answer
Horizontal asymptotes at y=0 and y=7
Y-intercept at (0,7/4)

24. Group Work: Graph and determine the horizontal asymptotes
𝐺(𝑥)= 𝑒 −4𝑥

25. Answer
Horizontal asymptotes y=0 and y=26
Y-intercept at (0,26/3)

26. Word Problems: Year 2000 782,248 people Year 2010 923,135 people
Use this information to determine when the population will surpass 1 million people? (hint use exponential function)

27. Group Work Year 1990 156,530 people Year 2000 531,365 people
Use this information and determine when the population will surpass 1.5 million people?

28. Word Problem The population of New York State can be modeled by
𝑓 𝑡 = 𝑒 − 𝑡
𝑓 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑚𝑖𝑙𝑙𝑖𝑜𝑛𝑠 𝑎𝑛𝑑 𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 1800
A) What’s the population in 1850?
B) What’s the population in 2010?
C) What’s the maximum sustainable population?

29. Answer
A) 1,794,558
B) 19,161,673
C) 19,875,000

30. Group Work
In chemistry, you are given half-life formulas 𝑃 𝑡 = 𝑃 0 𝑏 𝑡 𝑟 𝑟=ℎ𝑎𝑙𝑓 𝑙𝑖𝑓𝑒, 𝑡=𝑡𝑖𝑚𝑒, 𝑃 0 =𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 If you are given a certain chemical have a half-life of 56.3 minutes. If you are given 80 g first, when will it become 16 g?

31. Homework Practice
P 286 #1-54 eoe

32. Exponential and Logistic Modeling

33. Review
We learned that how to write exponential functions when given just data.
Now what if you are given other type of data? That would mean some manipulation

34. Quick Review

35. Quick Review Solutions

36. Exponential Population Model

37. Example: You are given 𝑃 𝑡 =300 1.05 𝑡
Is this a growth or decay? What is the rate?

38. Example Finding Growth and Decay Rates

39. Example You are given 𝑃 𝑡 =800 .15 𝑡
Is this a growth or decay? What is the rate?

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

41. Group Work
Suppose 50 bacteria is put into a petri dish and it doubles every hour. When will the bacteria be 350,000?

42. Answer
𝑃 𝑡 = 𝑡
350000= 𝑡
t=12.77 hours

43. Example Modeling Bacteria Growth

44. Group Work: half-life
Suppose the half-life of a certain radioactive substance is 20 days and there are 10g initially. Find the time when there will be 1 g of the substance.

45. answer
Just the setting up
𝑃 𝑡 =𝑃 𝑏 𝑡 ℎ𝑎𝑙𝑓𝑙𝑖𝑓𝑒
1= 𝑡 20

46. Group Work You are given 𝑃 𝑡 =150 1.025 𝑡
When will this become ?

47. Example Modeling U.S. Population Using Exponential Regression
Use the data and exponential regression to predict the U.S. population for 2003.

48. Example Modeling a Rumor

49. Example Modeling a Rumor: Answer

50. Key Word Maximum sustainable population
What does this mean? What function deals with this?

51. 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.

52. Homework Practice (Do in class also)
P 296 #1-44 eoo

53. Logarithmic Function, graphs and properties

54. Quick Review

55. Quick Review Solutions

56. Changing Between Logarithmic and Exponential Form

57. Group Work: transform logarithmic form into exponential form
B) 𝑙𝑜𝑔 = 1 2
C) 𝑙𝑜𝑔 2 60=𝑥
D) 𝑙𝑜𝑔𝑥=8

58. Group Work: convert exponential form into logarithmic form
5 𝑥 =34
5 −2 = 1 25
4 0 =1
16 1 =𝑝

59. Inverses of Exponential Functions

60. Basic Properties of Logarithms

61. An Exponential Function and Its Inverse

62. Common Logarithm – Base 10
Logarithms with base 10 are called common logarithms.
The common logarithm log10x = log x.
The common logarithm is the inverse of the exponential function y = 10x.

63. Basic Properties of Common Logarithms

64. Example Solving Simple Logarithmic Equations

65. Example Solving Simple Logarithmic Equations

66. Basic Properties of Natural Logarithms

67. Graphs of the Common and Natural Logarithm

68. Example Transforming Logarithmic Graphs

69. Example Transforming Logarithmic Graphs

70. Quick Review

71. Quick Review Solutions

72. 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.

73. Properties of Logarithms

74. Example Proving the Product Rule for Logarithms

75. Example Proving the Product Rule for Logarithms

76. Example Expanding the Logarithm of a Product

77. Example Expanding the Logarithm of a Product

78. Example Condensing a Logarithmic Expression

79. Example Condensing a Logarithmic Expression

80. Group Work
𝐸𝑥𝑝𝑎𝑛𝑑 log⁡(7 𝑥 2 𝑦 𝑧 5 )

81. Group Work
Expand
𝑙𝑜𝑔 𝑦 2 𝑥

82. Group Work
Express as a single logarithm
𝑙𝑜𝑔 𝑧 𝑡− 𝑙𝑜𝑔 𝑧 𝑥 +5 𝑙𝑜𝑔 𝑧 𝑚

83. Group Work Express as a single logarithm
4 𝑙𝑜𝑔 2 𝑥 𝑙𝑜𝑔 2 𝑦−3 𝑙𝑜𝑔 2 𝑧

84. Change-of-Base Formula for Logarithms

85. Example Evaluating Logarithms by Changing the Base

86. Example Evaluating Logarithms by Changing the Base

87. Solving
4 𝑥 =51

88. Solving
ln 𝑒

89. Solving
log 1

90. Solving
log 5𝑥= log 4+ log (𝑥−3)

91. Solving
𝑙𝑜𝑔 =𝑥

92. Solving
2 5+3𝑥 =16

93. Homework Practice
Pg 317 #1-50 eoe

94. Equation Solving and Modeling

95. Quick Review

96. Quick Review Solutions

97. One-to-One Properties

98. Example Solving an Exponential Equation Algebraically

99. Example Solving an Exponential Equation Algebraically

100. Example Solving a Logarithmic Equation

101. Example Solving a Logarithmic Equation

102. Group Work
ln 3𝑥−2 + ln 𝑥−1 =2𝑙𝑛𝑥

103. Group Work: Solve for x
𝑥 3 =5

104. Group Work: Solve
𝑒 𝑥 − 𝑒 −𝑥 2 =5

105. Group Work: Solve
1.05 𝑥 =8

106. 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.

107. Note:
In regular cases, how you determine the magnitude is by how many decimal places they differ
In term of Richter scale and pH level, since the number is the power or the exponent, you just take the difference of them.

108. Example:
What’s the difference of the magnitude between kilometer and meter?
It is 3 orders of magnitude longer than a meter

109. Example:
The order of magnitude between an earthquake rated 7 and Richter scale rated 5.5.
The difference of magnitude is 1.5

110. Group Work Find the order of magnitude: Between A dollar and a penny
A horse weighing 500 kg and a horse weighing 50g
8 million people vs population of 8

111. Answer 2 orders of magnitude 4 orders of magnitude

112. Group Work Find the difference of the magnitude:
Sour vinegar a pH of 2.4 and baking soda pH of 8.4
Earthquake in India 7.9 and Athens 5.9

113. Answer
6 orders of magnitude
2 orders of magnitude

114. Richter Scale

115. Example:
How many times more severe was the 2001 earthquake in Gujarat, India ( 𝑅 1 =7.9) than the 1999 earthquake in Athens, Greece ( 𝑅 2 =5.9)

116. Group Work: Show work
How many times more severs was the earthquake in SF ( 𝑅 1 =6.5) than the earthquake in PS ( 𝑅 2 =3.6)?

117. 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.

118. Example:
Sour vinegar has pH of 2.4 and a box of Leg and Sickle baking soda has a pH of 8.4.
A) what are their hydrogen-ion concentration?
B) How many more times greater is the hydrogen-ion concentration of the vinegar than of the baking soda?

119. Group Work A substance with pH of 3.4 and another with pH of 8.1
A) what are their hydrogen-ion concentration?
B) How many more times greater is the hydrogen-ion concentration?

120. Newton’s Law of Cooling

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

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

123. Group Work
A substance is at temperature 96℃ is placed in 16℃. Four minutes later the temperature of the egg is 45℃. Use Newton’s Law of Cooling to determine when the egg will be 20℃

124. Regression Models Related by Logarithmic Re-Expression
Linear regression: y = ax + b
Natural logarithmic regression: y = a + blnx
Exponential regression: y = a·bx
Power regression: y = a·xb

125. Three Types of Logarithmic Re-Expression

126. Three Types of Logarithmic Re-Expression (cont’d)

127. Three Types of Logarithmic Re-Expression (cont’d)

128. Homework Practice
Pg 331 #1-51 eoe

129. Mathematics of finance

130. Interest Compounded Annually

131. Interest Compounded k Times per Year

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

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

134. Group Work
Suppose you have $10000, you invest in a place where they give you 12% interest compounded quarterly. Find the value of your investment after 40 years.

135. Compound Interest – Value of an Investment

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

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

138. Group Work
Suppose you have $10000, you invest in a company where they give you 12% interest compounded continuously. Find the value of your investment after 40 years.

139. 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.

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

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

142. Future Value of an Annuity

143. Future Value of an Annuity
At the end of each quarter year, Emily makes a $500 payment into the Lanaghan Mutual Fund. If her investments earn 7.88% annual interest compounded quarterly, what will be the value of Emily’s annuity in 20 years?
Remember i=r/k

144. Group Work
You are currently 18 and you want to retire at age 65. You decide to invest in your future. You are putting in $35 month. If your investment earn 12% annual interest compounded monthly, what will the value of your annuity when you retire?

145. Present Value of an Annuity

146. Example
Mr. Liu bought a new car for $ What are the monthly payment for a 5 year loan with 0 down payment if the annual interest rate (APR) is 2.9%?

147. Homework Practice
Pg 341 #2-56 eoe