Chapter 7: Exponential and Logarithmic Functions 7.1 Exploring Exponential Models
y ab x
An exponential function is a function looks like:
a 0, b 0, b 1
Example 1: Graph the following exponential functions a) y 2
x
1 y 2
b)
y
x
y
x
x
c) y 2 3
x
1 y 3 4
d)
y
x
y
x
x
Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Exponential Behavior Two types of exponential behavior: Exponential Growth: as x increases, y increases Exponential Decay: as x increases, y decreases Exponential graphs are also asymptotic. An Asymptote is a line the graph approaches, but never crosses.
Example 2: Write D if the function represents exponential decay. Write G if it represents exponential growth. What is the y-intercept of the graph?
y-int = ______
y-int = ______
y-int = ______
y-int = ______
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Modeling Exponential Growth/Decay
You can model exponential growth/decay with the formula:
A(t ) a(1 r )t
A(t) =
Amount after “t” time periods (the ending value)
a=
initial amount (the starting value)
r=
the rate of growth (AS A DECIMAL!!!) positive for growth, negative for decay
t=
the number of time periods
Example 3: Draw a line from the function in Column A to the situation it models in Column B.
Example 4: Suppose you invest $500 in a savings account that pays 3.5% annual interest. How much will be in the account after five years?
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Example 4 continued…. Suppose you invest $500 in a savings account that pays 3.5% annual interest. When will the account contain $650?
Example 5: Suppose you deposit $3000 in a savings account that pays interest at an annual rate of 4%. a) If no money is added or withdrawn from the account, how much will be in the account after 10 years?
b) When will the account be worth $5000?
Example 6: Write an exponential model for each situation. Find the amount after the specified amount of time. a) A population of birds is currently at 200,000 birds. The population is decreasing at a rate of 3.5% per year. How many birds will there be in 50 years?
b) An investment of $75,000 increases at a rate of 12.5% per year. What is the value of the investment after 30 years?
c) A new truck that sells for $29,000 depreciates 12% per year. What is the value of the truck after 7 years?
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
5
7.2 Properties of Exponential Functions Example1: Graph y 2 x
Graph y 2 x 3
Example 2:
y
x
Example 3: Graph y 2 x 5
y
y
Example 4: Graph y 2(2) x
x
y
x
x
Each of Ex. 2-4 represent a transformation of the parent function y 2 x We talked about transformations in Lesson 2.6 last semester.
Algebra 2B: Chapter 7 Notes Recall from 2.6
Example 5: How does the graph of y 4 x 2 compare to the graph of the parent function y 4 x ?
The graph at the right shows y 4 x . Sketch the graph of y 4 x 2 on the same set of axes.
Example 6: How does the graph of y 4 x 2 compare to the graph of the parent function y 4 x ?
The graph at the right shows y 4 x . Sketch the graph of y 4 x 2 on the same set of axes
Exponential and Logarithmic Functions
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Algebra 2B: Chapter 7 Notes Example 7:
Exponential and Logarithmic Functions
State the parent function. Describe how the graph of each function is a transformation of the parent function. x
a)
c)
1 y 2 6 3
b)
y
1 x 5 3 4
Parent:
Parent:
Transformations:
Transformations:
y 5
x 6
8
d)
y 2 1 x
Parent:
Parent:
Transformations:
Transformations:
Natural Base Exponentials: Base “e”
e 2.72
called the “natural base” or “Euler’s number” used to describe continuous growth or decay has the same exponential properties as any other exponential base
You can evaluate powers of “e” in your calculator using the “e^x” button. This is usually a shift command of the button called “LN”. Example 8: Estimate to four decimal places. a)
e
4
b)
e
2
c)
e
3 4
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
In 7.1, we talked about interest compounded annually. For continuously compounded interest, we use the number “e”. Modeling Continuous Exponential Growth/Decay
You can model exponential growth/decay with the formula:
A(t ) P e r t
A(t) =
Amount after “t” time periods (the ending value)
P=
initial amount (the starting value)
r=
the annual rate of growth (AS A DECIMAL!!!) positive for growth, negative for decay
t=
time in years
Example 9: Suppose you won a contest at the start of 9th grade that deposited $3000 in an account that pays 5% annual interest compounded continuously. About how much will be in the account after you graduate from college in 8 years?
Example 10: Find the amount in a continuously compounded account for the given conditions. principal: $600 annual interest rate: 4.5% time: 5 years
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
7.3 Logarithmic Functions as Inverses A LOGARITHM is an EXPONENT!!! 52 = 25
The base is _____ . The exponent is_____.
When b 0 and b 1
We say “The log base _____ of _____ is _____.”
bx n
We write: 491/2 =7
if and only if
The base is _____ . The exponent is _____. We say “The log base _____ of ____ is _____.”
logb n x
We write: 103=1000
The base is _____. The exponent is _____. We say “The log base _____ of _____ is _______.”
The logarithm is the exponent you must use on base “b” to get the number “n”
We write:
Example 1: What is the logarithmic form of each equation? 3
a)
6 36 2
8 2 b) 3 27
0 c) 1 3
Example 2: What is the exponential form of each equation? a) log 3 81 4
b) log16 8 0.75
c) 5 log 2
1 32
A common logarithm is a log base 10. You can write log10 x log x , without showing the 10. Ex.
102 100
Ex.
log 40 1.6
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Evaluating Logarithms
Exponential Property of Equality
What is the value of log 5 125 ?
For b 0 and b 1,
This question is asking you “What EXPONENT do I need on base 5 to get 125?” Sometimes in problems like this, it is easy to “see” what the answer should be. When it isn’t so easy, follow these steps.
If
bx b y
Then
x y
Step 1: Write the logarithm. Set equal to x, or some other variable. Step 2: Rewrite the logarithm as an exponential. Step 3: Rewrite both sides of the equation to use the same base. Step 4: Use the property of equality (box upper right) to set exponents equal. Step 5: Solve for your variable.
Example 3: Evaluate each logarithm. a)
log 4 64
b)
log 2 32
d)
log 8 32
e)
log 7
1 49
c)
log 9 27
f)
log125
1 25
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Algebra 2B: Chapter 7 Notes
y Exponential and Logarithmic Functions
11
Logarithms and exponential functions are inverses of each other.
Example 4: Graph y 2 x . Then use that graph to draw the graph of y log 2 x
x
Transformations work with logarithmic functions, too.
Example 5: How does the graph of the logarithmic function compare to the graph of the parent function? a) y log 2 ( x 3) 4 (see graph above)
b)
y 5log 2 x
c) y log( x 4) 5
Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
7.4 Properties of Logarithms A logarithm is an exponent! Therefore, the properties of logarithms corresponds to the properties of exponents.
Example 1: What is each expression written as a single logarithm? Simplify if possible. a) log 3 9 log 3 24
b) log 2 7 log 2 9
c) log 4 163
d) 3log 7 x 5log 7 y
e) 4log x 3log x
f) log 2 16 log 2 8
g) log 4 5 x log 4 3 x
h)
1 2log 4 6 log 4 9 log 4 27 3
i) log 2 85
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Your calculator can only estimate common logarithms (log base 10). Example 2: Use your calculator to estimate the following to four decimal places. a) log1000
b) log 45
c) log 0.003
To calculate a logarithm that is NOT base 10, you can use the change of base formula.
Example 3: Use the change of base formula to evaluate each expression. Round your answer to four decimal places. a) log 2 32
b) log 4 70
c) log8 1000
7.5 Exponential and Logarithmic Equations Solving Exponential Equations Exponential Property of Equality
With Common Bases: Example 1:
2x6 24 x
Example 2:
273 x 81
For b 0 and b 1, If
bx b y
Then
x y
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Not Common Bases: Example 3:
Step 1: Isolate the exponential.
5x 130
Step 2: Take the common log of both sides of the equation. Step 3: Use the power property for logarithms to bring down the exponent. Step 4: Solve the equation for x. Step 5: Use your calculator to round your answer to four decimal places
Example 4:
73 x 24 184
Example 5:
25 x 4 7
Logarithmic Equations One Log in the Equation Example 6:
log(2 x 4) 3
Step 1: Rewrite the logarithm as an exponential. Step 2: Simplify and Solve Step 3: Check your answer.
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Algebra 2B: Chapter 7 Notes Example 7:
log 4 x 3 2
Exponential and Logarithmic Functions Example 8:
Multiple Logs in an Equation Example 9:
log(2 x 8) 1
Step 1: Combine the logarithms using the properties of logarithms.
2log x log 4 3
Step 2: Rewrite as an exponential. Step 3: Simplify and Solve. Step 4: Check your answer.
Example 10:
log( x 3) log 4 2
Example 11:
log10 log 2 x 3
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
7.6 Natural Logarithms Recall that the inverse of an exponential is a logarithm. Inverses
2 x and log 2 x
x The “natural exponential” e also has an inverse:
3x and log 3 x 10 x and log x
Natural logarithm
log e x ln x
Example 1: Rewrite in the inverse format exponential logarithmic or logarithmic exponential a)
e2 x
b)
e x 27
c)
ln x 3
The properties of logarithms still apply to natural logarithms as well. Example 2: What is each expression written as a single logarithm? a) ln 7 2ln5
c)
3ln x 2ln y ln 5
b) 3ln x 2ln 2 x
d)
ln 4 x
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Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Example 3: Solve. Round your answer to four decimal places as necessary. a)
