1. Exponential Functions
SECTION 4.1
Exponential Functions
Define an exponential function.
Graph exponential functions.
Use transformations on exponential functions.
Define simple interest.
Develop a compound interest formula. Understand the number e. 1 2 3 4 5 6
Exponential Functions

2. EXPONENTIAL FUNCTION A function f of the form
is called an exponential function with base a. Its domain is (–∞, ∞).

3. d. Let F(x) = 4x. Find F(3.2). EXAMPLE 1
Evaluating Exponential Functions
d. Let F(x) = 4x. Find F(3.2).

4. d. F(3.2) = 43.2 ≈ 84.44850629 Solution EXAMPLE 1
Evaluating Exponential Functions
Solution
d. F(3.2) = ≈

5. RULES OF EXPONENTS
Let a, b, x, and y be real numbers with a > 0 and b > 0. Then

6. Graph the exponential function
Graphing an Exponential Function with Base a > 1 – Exponential Growth
EXAMPLE 2
Graph the exponential function
Solution
Make a table of values.
Plot the points and draw a smooth curve.

7. This graph is typical for exponential functions when a > 1.
Graphing an Exponential Function with Base a > 1
EXAMPLE 2
Solution continued
This graph is typical for exponential functions when a > 1.

8. Plot the points and draw a smooth curve.
Graphing an Exponential Function with Base 0 < a < 1 – Exponential Decay
EXAMPLE 3
Sketch the graph of
Solution
Make a table of values.
Plot the points and draw a smooth curve.

9. As x increases in the positive direction, y decreases towards 0.
Graphing an Exponential Function with Base 0 < a < 1
EXAMPLE 3
Solution continued
As x increases in the positive direction, y decreases towards 0.

10. PROPERTIES OF EXPONENTIAL FUNCTIONS
Let f (x) = ax, a > 0, a ≠ 1.
1. The domain of f (x) = ax is (–∞, ∞).
2. The range of f (x) = ax is (0, ∞); the entire graph lies above the x-axis.
3. For a > 1, Exponential Growth
(i) f is an increasing function, so the graph rises to the right.
(ii) as x → ∞, y → ∞.
(iii) as x → –∞, y → 0.

11. 4. For 0 < a < 1, - Exponential Decay
(i) f is a decreasing function, so the graph falls to the right.
(ii) as x → – ∞, y → ∞.
(iii) as x → ∞, y → 0.
5. The graph of f (x) = ax has no x-intercepts, so it never crosses the x-axis. No value of x will cause f (x) = ax to equal 0.
6. The graph of is a smooth and continuous curve, and it passes through the points
7. The x-axis is a horizontal asymptote for every exponential function of the form f (x) = ax.

12. TRANSFORMATIONS ON EXPONENTIAL FUNCTION f (x) = ax
Equation
Effect on Equation
Horizontal
Shift
y = ax+b
Shift the graph of y = ax, |b| units
left if b > 0.
right if b < 0.
Vertical
Shift
y = ax + b
Shift the graph of y = ax, |b| units
up if b > 0.
down if b < 0.

13. TRANSFORMATIONS ON EXPONENTIAL FUNCTION f (x) = ax
Equation
Effect on Equation
Stretching or Compressing
(Vertically)
y = cax
Multiply the y coordinates by c. The graph of y = ax is vertically
stretched if c > 1.
compressed if
0 < c < 1.

14. TRANSFORMATIONS ON EXPONENTIAL FUNCTION f (x) = ax
Equation
Effect on Equation
Reflection
y = –ax
Reflect the graph of y = ax in the
x-axis.
y = a–x
Reflect the graph of y = ax in the
y-axis.

15. Use transformations to sketch the graph of each function.
EXAMPLE 6
Sketching Graphs
Use transformations to sketch the graph of each function.
State the domain and range of each function and the horizontal asymptote of its graph.

16. Horizontal Asymptote: y = –4
EXAMPLE 6
Sketching Graphs
Solution
Domain: (–∞, ∞)
Range: (–4, ∞)
Horizontal Asymptote: y = –4

17. Horizontal Asymptote: y = 0
EXAMPLE 6
Sketching Graphs
Solution continued
Domain: (–∞, ∞)
Range: (0, ∞)
Horizontal Asymptote: y = 0

18. Horizontal Asymptote: y = 0
EXAMPLE 6
Sketching Graphs
Solution continued
Domain: (–∞, ∞)
Range: (–∞, 0)
Horizontal Asymptote: y = 0

19. Horizontal Asymptote: y = 2
EXAMPLE 6
Sketching Graphs
Solution continued
Domain: (–∞, ∞)
Range: (–∞, 2)
Horizontal Asymptote: y = 2

20. General Exponential Growth/Decay Model
Rate of decay (r < 0),Growth (r > 0)
Amount after t time periods
Number of time periods
Original amount

21. COMPOUND INTEREST – Growth
Compound interest is the interest paid on both the principal and the accrued (previously earned) interest. It is an application of exponential growth.
Interest that is compounded annually is paid once a year. For interest compounded annually, the amount A in the account after t years is given by
Rate of decay (r < 0),Growth (r > 0)
Amount after t time periods
Number of time periods
Original amount 21

22. How much money will she have in her account after five years?
EXAMPLE 2
Calculating Compound Interest
Juanita deposits $8000 in a bank at the interest rate of 6% compounded annually for five years.
How much money will she have in her account after five years?
b. How much interest will she receive? 22

23. a. Here P = $8000, r = 0.06, and t = 5. b. Interest = A  P
EXAMPLE 2
Calculating Compound Interest
Solution
a. Here P = $8000, r = 0.06, and t = 5.
b. Interest = A  P
= $10,  $8000 = $ 23

24. COMPOUND INTEREST FORMULA
A = amount after t years
P = principal
r = annual interest rate (expressed as a decimal)
n = number of times interest is compounded each year
t = number of years 24

25. Using Different Compounding Periods to Compare Future Values
EXAMPLE 3
If $100 is deposited in a bank that pays 5% annual interest, find the future value A after one year if the interest is compounded
annually.
semiannually.
quarterly.
monthly.
daily. 25

26. (i) Annual Compounding:
Using Different Compounding Periods to Compare Future Values
EXAMPLE 3
Solution
In the following computations, P = 100, r = 0.05 and t = 1. Only n, the number of times interest is compounded each year, changes. Since t = 1, nt = n(1) = n.
(i) Annual Compounding: 26

27. (ii) Semiannual Compounding:
Using Different Compounding Periods to Compare Future Values
EXAMPLE 3
(ii) Semiannual Compounding:
(iii) Quarterly Compounding: 27

28. (iv) Monthly Compounding:
Using Different Compounding Periods to Compare Future Values
EXAMPLE 3
(iv) Monthly Compounding:
(v) Daily Compounding: 28

29. the initial number of bacteria,
EXAMPLE 8
Bacterial Growth
A technician to the French microbiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubles every hour. If the bacteria count B(t) is modeled by the equation
with t in hours, find
the initial number of bacteria,
the number of bacteria after 10 hours; and
the time when the number of bacteria will be 32,000.

30. After 4 hours, the number of bacteria will be 32,000.
EXAMPLE 8
Bacterial Growth
Solution
a. Initial size
c. Find t when B(t) = 32,000
After 4 hours, the number of bacteria will be 32,000.

31. THE VALUE OF e As h gets larger and larger,
gets closer and closer to a fixed number. This irrational number is denoted by e and is sometimes called the Euler number.
The value of e to 15 places is
e = 31

32. CONTINUOUS COMPOUND FORMULA
A = amount after t years
P = principal
r = annual rate (expressed as a decimal)
t = number of years 32

33. EXAMPLE 4
Calculating Continuous Compound Interest
Find the amount when a principal of $8300 is invested at a 7.5% annual rate of interest compounded continuously for eight years and three months.
Solution
P = $8300 and r = Convert eight years and three months to 8.25 years. 33

34. EXAMPLE 5
Calculating the Amount of Repaying a Loan
How much money did the government owe DeHaven’s descendants for 213 years on a $450,000 loan at the interest rate of 6%?
Solution
a. With simple interest, 34

35. b. With interest compounded yearly,
EXAMPLE 5
Calculating the Amount of Repaying a Loan
Solution continued
b. With interest compounded yearly,
c. With interest compounded quarterly, 35

36. d. With interest compounded continuously,
EXAMPLE 5
Calculating the Amount of Repaying a Loan
Solution continued
d. With interest compounded continuously,
Notice the dramatic difference between quarterly and continuous compounding and the dramatic difference between simple interest and compound interest. 36

37. THE NATURAL EXPONENTIAL FUNCTION
The exponential function
with base e is so prevalent in the sciences that it is often referred to as the exponential function or the natural exponential function. 37

38. Use transformations to sketch the graph of
EXAMPLE 6
Sketching a Graph
Use transformations to sketch the graph of
Solution
Start with the graph of y = ex. 38

39. Use transformations to sketch the graph of
EXAMPLE 6
Sketching a Graph
Use transformations to sketch the graph of
Solution coninued
Shift the graph of y = ex one unit right. 39

40. Use transformations to sketch the graph of
EXAMPLE 6
Sketching a Graph
Use transformations to sketch the graph of
Solution continued
Shift the graph of y = ex – 1 two units up. 40

41. MODEL FOR EXPONENTIAL GROWTH OR DECAY
A(t) = amount at time t
A0 = A(0), the initial amount
k = relative rate of growth (k > 0) or decay (k < 0)
t = time 41

42. EXAMPLE 7
Modeling Exponential Growth and Decay
In the year 2000, the human population of the world was approximately 6 billion and the annual rate of growth was about 2.1%. Using the model on the previous slide, estimate the population of the world in the following years.
2030
1990 42

43. EXAMPLE 7
Modeling Exponential Growth and Decay
Solution
a. The year 2000 corresponds to t = 0. So A0 = 6 (billion), k = 0.021, and 2030 corresponds to t = 30.
The model predicts that if the rate of growth is 2.1% per year, over billion people will be in the world in 2030. 43

44. b. The year 1990 corresponds to t = 10.
EXAMPLE 7
Modeling Exponential Growth and Decay
Solution
b. The year 1990 corresponds to t = 10.
The model predicts that the world had over 4.86 billion people in (The actual population in 1990 was 5.28 billion.) 44

45. Logarithmic Functions
SECTION 4.3
Logarithmic Functions 1
Define logarithmic functions.
Inverse Functions
Evaluate logarithms.
Rules of Logarithms
Find the domains of logarithmic functions.
Graph logarithmic functions.
Use logarithms to evaluate exponential equations. 2 3 4 5 6 7 45

46. DEFINITION OF THE LOGARITHMIC FUNCTION
For x > 0, a > 0, and a ≠ 1,
The function f (x) = loga x, is called the logarithmic function with base a.
The logarithmic function is the inverse function of the exponential function. 46

47. Inverse Functions
Certain pairs of one-to-one functions “undo” one another. For example, if
then

48. Inverse Functions
Starting with 10, we “applied” function  and then “applied” function g to the result, which returned the number 10.

49. Inverse Functions
As further examples, check that

50. Inverse Functions In particular, for this pair of functions,
In fact, for any value of x, or
Because of this property, g is called the inverse of .

51. Let  be a one-to-one function. Then g is the inverse function of  if
for every x in the domain of g,
and
for every x in the domain of .

52. Write each exponential equation in logarithmic form.
Converting from Exponential to Logarithmic Form
EXAMPLE 1
Write each exponential equation in logarithmic form.
Solution 52

53. Write each logarithmic equation in exponential form.
Converting from Logarithmic Form to Exponential Form
EXAMPLE 2
Write each logarithmic equation in exponential form.
Solution 53

54. Find the value of each of the following logarithms.
EXAMPLE 3
Evaluating Logarithms
Find the value of each of the following logarithms.
Solution 54

55. EXAMPLE 3
Evaluating Logarithms
Solution continued 55

56. Solve each equation. Solution EXAMPLE 4
Using the Definition of Logarithm
Solve each equation.
Solution 56

57. EXAMPLE 4
Using the Definition of Logarithm
Solution continued 57

58. EXAMPLE 4
Using the Definition of Logarithm
Solution continued 58

59. Rules of Logarithms Rules of Logarithms with Base a
If M, N, and a are positive real numbers with a ≠ 1, and x is
any real number, then
1. loga(a) = loga(1) = 0
3. loga(ax) = x 4.
5. loga(MN) = loga(M) + loga(N)
6. loga(M/N) = loga(M) – loga(N)
7. loga(Mx) = x · loga(M) 8. loga(1/N) = – loga(N)
These relationships are used to solve exponential or logarithmic equations

60. COMMON LOGARITHMS
The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x = log10 x. Thus,
y = log x if and only if x = 10 y.
Applying the basic properties of logarithms
log 10 = 1
log 1 = 0
log 10x = x 60

61. NATURAL LOGARITHMS
The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = loge x. Thus,
y = ln x if and only if x = e y.
Applying the basic properties of logarithms
ln e = 1
ln 1 = 0
log ex = x 61

62. DOMAIN OF LOGARITHMIC FUNCTION
Domain of y = loga x is (0, ∞)
Range of y = loga x is (–∞, ∞)
Logarithms of 0 and negative numbers are not defined. 62

63. Domain of a logarithmic function must be positive, that is,
EXAMPLE 5
Finding the Domain
Find the domain of
Solution
Domain of a logarithmic function must be positive, that is,
The domain of f is (–∞, 2). 63

64. Sketch the graph of y = log3 x.
EXAMPLE 6
Sketching a Graph
Sketch the graph of y = log3 x.
Solution by plotting points (Method 1)
Make a table of values. 64

65. EXAMPLE 6
Sketching a Graph
Solution continued
Plot the ordered pairs and connect with a smooth curve to obtain the graph of y = log3 x. 65

66. EXAMPLE 6
Sketching a Graph
Solution by using the inverse function (Method 2)
Graph y = f (x) = 3x.
Reflect the graph of y = 3x in the line y = x to obtain the graph of y = f –1(x) = log3 x. 66 66

67. GRAPHS OF LOGARITHMIC FUNCTIONS67

68. PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Exponential Function f (x) = ax
Logarithmic Function f (x) = loga x
Domain (–∞, ∞)
Range (0, ∞)
Domain (0, ∞) Range (–∞, ∞)
2. y-intercept is 1
No x-intercept
x-intercept is No y-intercept
3. x-axis (y = 0) is the horizontal asymptote
y-axis (x = 0) is the vertical asymptote 68

69. PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Exponential Function f (x) = ax
Logarithmic Function f (x) = loga x
The graph is a continuous smooth curve that passes through the points
(0, 1), and (1, a).
The graph is a continuous smooth
curve that passes through the points
(1, 0), and (a, 1). 69

70. PROPERTIES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Exponential Function f (x) = ax
Logarithmic Function f (x) = loga x
5. Is one-to-one , that
is, au = av if and
only if u = v.
Is one-to-one, that is, logau = logav if and only if u = v.
Increasing if a > 1
Decreasing if 0 < a < 1
Increasing if a > 1 Decreasing if 0 < a < 1 70

71. EXAMPLE 7
Using Transformations
Start with the graph of f (x) = log3 x and use transformations to sketch the graph of each function.
State the domain and range and the vertical asymptote for the graph of each function. 71

72. Shift up 2 Domain (0, ∞) Range (–∞, ∞) Vertical asymptote x = 0
EXAMPLE 7
Using Transformations
Solution
Shift up 2
Domain (0, ∞)
Range (–∞, ∞)
Vertical asymptote x = 0 72

73. Shift right 1 Domain (1, ∞) Range (–∞, ∞) Vertical asymptote x = 1
EXAMPLE 7
Using Transformations
Solution continued
Shift right 1
Domain (1, ∞)
Range (–∞, ∞)
Vertical asymptote x = 1 73

74. Reflect graph of y = log3 x in the x-axis Domain (0, ∞)
EXAMPLE 7
Using Transformations
Solution continued
Reflect graph of y = log3 x in the x-axis Domain (0, ∞)
Range (–∞, ∞)
Vertical asymptote x = 0 74

75. Reflect graph of y = log3 x in the y-axis Domain (∞, 0)
EXAMPLE 7
Using Transformations
Solution continued
Reflect graph of y = log3 x in the y-axis Domain (∞, 0)
Range (–∞, ∞)
Vertical asymptote x = 0 75

76. Start with the graph of f (x) = log x.
EXAMPLE 8
Using Transformations to Sketch a Graph
Sketch the graph of
Solution
Start with the graph of f (x) = log x.
Step 1: Replacing x with x – 2 shifts the
graph two units right. 76

77. Step 3: Adding 2 shifts the graph two units up.
EXAMPLE 8
Using Transformations to Sketch a Graph
Solution continued
Step 2: Multiplying
by 1 reflects the graph
Step 3: Adding 2 shifts the graph two units up.
in the x-axis. 77

78. Rules of Logarithms Rules of Logarithms with Base a
If M, N, and a are positive real numbers with a ≠ 1, and x is
any real number, then
1. loga(a) = loga(1) = 0
3. loga(ax) = x 4.
5. loga(MN) = loga(M) + loga(N)
6. loga(M/N) = loga(M) – loga(N)
7. loga(Mx) = x · loga(M) 8. loga(1/N) = – loga(N)

79. Given that log 5 z = 3 and log 5 y = 2, evaluate each expression.
Using Rules of Logarithms to Evaluate Expressions
EXAMPLE 1
Given that log 5 z = 3 and log 5 y = 2, evaluate each expression.
Solution 79

80. Solution continued Using Rules of Logarithms to Evaluate Expressions
EXAMPLE 1
Solution continued 80

81. Solution continued Using Rules of Logarithms to Evaluate Expressions
EXAMPLE 1
Solution continued 81

82. Write each expression in expanded form.
EXAMPLE 2
Writing Expressions In Expanded Form
Write each expression in expanded form.
Solution 82

83. EXAMPLE 2
Writing Expressions In Expanded Form
Solution continued 83

84. Write each expression in condensed form.
EXAMPLE 3
Writing Expressions in Condensed Form
Write each expression in condensed form. 84

85. EXAMPLE 3
Writing Expressions in Condensed Form
Solution 85

86. EXAMPLE 3
Writing Expressions in Condensed Form
Solution continued 86

87. EXAMPLE 3
Writing Expressions in Condensed Form
Solution continued 87

88. CHANGE-OF-BASE FORMULA
Let a, b, and x be positive real numbers with a ≠ 1 and b ≠ 1. Then logb x can be converted to a different base as follows: 88

89. Using a Change of Base to Compute Logarithms
EXAMPLE 4
Compute log513 by changing to a. common logarithms and b. natural logarithms.
Solution 89

90. Evaluate each expression.
EXAMPLE 9
Evaluating the Natural Logarithm
Evaluate each expression.
Solution
Use a calculator. 90

91. If P is the original amount invested, A = 2P.
EXAMPLE 10
Doubling Your Money
How long will it take to double your money if it earns 6.5% compounded continuously?
b. At what rate of return, compounded continuously, would your money double in 5 years?
Solution
If P is the original amount invested,
A = 2P.
It will take 11 years to double your money. 91

92. Your investment will double in 5 years at the rate of 13.86%.
EXAMPLE 10
Doubling Your Money
Solution continued b.
Your investment will double in 5 years at the rate of 13.86%. 92

93. Solving Exponential Or Logarithmic Equations
To solve an exponential or logarithmic equation, change the given equation into one of the following forms, where a and b are real numbers, a > 0 and
a ≠ 1, and follow the guidelines.
ax = b
Solve by taking logarithms on both sides.
2. Loga x = b
Solve by changing to exponential form ab = x.

94. Solve 7x = 12. Give the solution to the nearest thousandth.
SOLVING AN EXPONENTIAL EQUATION
Solve 7x = 12. Give the solution to the nearest thousandth.
Solution
While any appropriate base b can be used, the best practical base is base 10 or base e. We choose base e (natural) logarithms here.

95. Solve 7x = 12. Give the solution to the nearest thousandth.
SOLVING AN EXPONENTIAL EQUATION
Solve 7x = 12. Give the solution to the nearest thousandth.
Solution
Property of logarithms
Power of logarithms
Divide by In 7.
Use a calculator.
The solution set is {1.277}.

96. Solution SOLVING AN EXPONENTIAL EQUATION
Solve 32x – 1 = .4x+2 . Give the solution to the nearest thousandth.
Solution
Take natural logarithms on both sides.
Property power
Distributive property

97. Solution SOLVING AN EXPONENTIAL EQUATION
Solve 32x – 1 =.4x+2 . Give the solution to the nearest thousandth.
Solution
Write the terms with x on one side
Factor out x.
Divide by
2 In 3 – In .4.
Power property

98. Solution SOLVING AN EXPONENTIAL EQUATION
Solve 32x – 1 =.4x+2 . Give the solution to the nearest thousandth.
Solution
Apply the exponents.
This is exact.
Product property; Quotient property
This is approximate.
The solution set is { –.236}.

99. a. Solution SOLVING BASE e EXPONENTIAL EQUATIONS
Solve the equation. Give solutions to the nearest thousandth. a.
Solution
Take natural logarithms on both sides.
In = x2

100. a. Solution SOLVING BASE e EXPONENTIAL EQUATIONS
Solve the equation. Give solutions to the nearest thousandth. a.
Remember both roots.
Solution
Square root property
Use a calculator.
The solution set is { 2.302}.

101. b. Solution SOLVING BASE e EXPONENTIAL EQUATIONS
Solve the equation. Give solutions to the nearest thousandth. b.
Solution
Divide by e;
Take natural logarithms on both sides.
Power property

102. b. Solution SOLVING BASE e EXPONENTIAL EQUATIONS
Solve the equation. Give solutions to the nearest thousandth. b.
Solution
In e = 1
Multiply by – ½
The solution set is {– .549}.

103. Solution SOLVING A LOGARITHMIC EQUATION
Solve log(x + 6) – log(x + 2) = log x.
Solution
Quotient property
Property of logarithms

104. Solution SOLVING A LOGARITHMIC EQUATION
Solve log(x + 6) – log(x + 2) = log x.
Solution
Distributive property
Standard form
Factor.
Zero-factor property
The proposed negative solution (x = – 3) is not in the domain of the log x in the original equation, so the only valid solution is the positive number 2, giving the solution set {2}.

105. Solution SOLVING A LOGARITHMIC EQUATION
Solve log(3x + 2) + log(x – 1 ) = 1. Give the exact value(s) of the solution(s).
Solution
Substitute.
Product property
Property of logarithms

106. Solution SOLVING A LOGARITMIC EQUATION
Solve log(3x + 2) + log(x – 1 ) = 1. Give the exact value(s) of the solution(s).
Solution
Multiply.
Subtract 10.
Quadratic formula

107. Solution SOLVING A LOGARITMIC EQUATION
Solve log(3x + 2) + log(x – 1 ) = 1. Give the exact value(s) of the solution(s).
Solution
The number is negative, so x – 1 is negative. Therefore, log(x – 1) is not defined and this proposed solution must be discarded.
Since > 1, both 3x + 2 and x – 1 are positive and the solution set is

108. NEWTON’S LAW OF COOLING
Newton’s Law of Cooling states that
where T is the temperature of the object at time t, Ts is the surrounding temperature, and T0 is the value of T at t = 0.
108

109. EXAMPLE 11
McDonald’s Hot Coffee
The local McDonald’s franchise has discovered that when coffee is poured from a
coffeemaker whose contents are 180ºF into a noninsulated pot, after 1 minute, the coffee cools to 165ºF if the room temperature is 72ºF. How long should the employees wait before pouring the coffee from this noninsulated pot into cups to deliver it to customers at 125ºF?
109

110. Use Newton’s Law of Cooling with T0 = 180 and Ts = 72 to obtain
EXAMPLE 11
McDonald’s Hot Coffee
Solution
Use Newton’s Law of Cooling with T0 = 180 and Ts = 72 to obtain
We have T = 165 and t = 1.
110

111. Substitute this value for k.
EXAMPLE 11
McDonald’s Hot Coffee
Solution continued
Substitute this value for k.
Solve for t when T = 125.
The employee should wait about 5 minutes.
111

112. GROWTH AND DECAY MODEL A is the quantity after time t.
A0 is the initial (original) quantity (when t = 0).
r is the growth or decay rate per period.
t is the time elapsed from t = 0.
112

113. EXAMPLE 12
Chemical Toxins in a Lake
A chemical spill deposits 60,000 cubic meters of soluble toxic waste into a large lake. If 20% of the waste is removed every year, how many years will it take to reduce the toxin to 1000 cubic meters?
Solution
In the equation A = A0ert, we need to find A0, r, and the time when A = 1000.
113

114. Find A0. Initially (t = 0), we are given A0 = 60,000. So
EXAMPLE 12
Chemical Toxins in a Lake
Solution continued
Find A0. Initially (t = 0), we are given
A0 = 60,000. So
Find r. When t = 1 year, the amount of toxin
will be 80% of its initial value, or
114

115. continued So Solution continued EXAMPLE 12 Chemical Toxins in a Lake
115

116. It will take approximately 18 years to reduce toxin to 1000 m3.
EXAMPLE 12
Chemical Toxins in a Lake
Solution continued
3. Find t when A = 1000.
It will take approximately 18 years to reduce toxin to 1000 m3.
116