Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
1
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 = ______
2
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?
3
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a) If no money is added or withdrawn from the account, how much will be in the account after 10 years?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?
4
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
6
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
7
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  er 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
8
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)   
27
3
0
c) 1  3
Example 2: What is the exponential form of each equation?
a) log3 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
9
Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Evaluating Logarithms
Exponential Property of
Equality
What is the value of log5 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)
log8 32
e)
log 7
1
49
c)
log 9 27
f)
log125
1
25
10
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) log3 9  log3 24
b) log 2 7  log 2 9
c) log 4 163
d) 3log7 x  5log 7 y
e) 4 log x  3log x
f) log 2 16  log 2 8
g) log 4 5 x  log 4 3x
h)
1
2 log 4 6  log 4 9  log 4 27
3
i) log 2 85
12
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:
2x6  24 x
Example 2:
273 x  81
For b  0 and b  1,
If
bx  b y
Then
x y
13
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:
Step 1: Rewrite the logarithm as an
exponential.
log(2 x  4)  3
Step 2: Simplify and Solve
Step 3: Check your answer.
14
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.
2 log 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
15
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  2 ln y  ln 5
b) 3ln x  2ln 2x
d)
ln 4  x
16
Algebra 2B: Chapter 7 Notes
Exponential and Logarithmic Functions
Example 3: Solve. Round your answer to four decimal places as necessary.
a)
ln x  2
c) ln 2x  ln3  2
e)
2e x  20
2
b) ln(3x  5)  4
d)
e x  2  12
3x
f) e  5  15
17