Logarithmic and Exponential Functions By: Heather McGuire 3/21/08
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Logarithmic and Exponential Functions
By: Heather McGuire3/21/08
The following is a review for students before they take the chapter test on exponential and logarithmic functions. They will be able to test their knowledge on the various topics and hopefully have feedback to help them study for the exam. Students will be able to work at their own pace.
Cobb County Algebra 2 Standards
M.ALGII.3.5 Logarithmic: Solve/Equations The learner will be able to determine value of common and the natural logarithms and antilogarithms using a calculator. Applies the change of base rule.
M.ALGII.3.8 Logarithms: Definition/Properties The learner will be able to apply the definition and properties of logarithms to evaluate logarithms. Recognize and apply the inverse relationship of logarithms and exponential functions and graphs each function.
M.ALGII.4.4 Predictions: Regression Techniques The learner will be able to solve exponential and logarithmic equations, such as those involving growth, decay, and compound interest. Make predictions from collected data by applying regression techniques.
M.ALGII.4.8 Problem Solving: Relate The learner will be able to solve problems that relate concepts to other concepts and to practical applications using tools such as scientific or graphing calculators and computers.
The number eReview
2.718281828459eThe rules of exponents we have learned in the last 2
chapters also apply to e. Click the sound buttons to here the descriptions.
2 3 5
n m n me e e
e e e
5 2
3
2
4 2
nn m
m
ee
e
e e
e
43 4 3 4
12
2 2
16
mn n me e
e e
e
Now it is your turn to practice.
Work these problems on a separate sheet of paper. Then click on the answer button to see the answer.
Click the arrow at the bottom when you are ready to proceed to the next page.
7 5a. 12 2e e
8
10
4eb.
2e
Answer for a.
Answer for b.
24xc. 3e
Answer for c.
Answer for a
7 5a. 12 2e e
7 524 e e
First you multiply the numbers.
Then add the exponents of e.
224e
Click the u-turn arrow to return to the problem page.
Congratulations you did it.
Click the u-turn arrow to return to the problem page.
8
10
4eb.
2e
Answer for b
First you divide the numbers.
8
10
2eb.
eThen subtract the exponents of e and place e where the largest exponent was.
2
2b.
eCongratulations you did it.
Click the u-turn arrow to return to the problem page.
24xc. 3e
Answer for c
First you distribute the power to everything inside the parentheses.
Then use the rule of negative exponents to move everything.
Congratulations you did it.
2 8c. 3 xe
8x
1c.
9e
LogarithmsReview
The only logarithms you can calculate on the calculator are base 10, the common
log, and base e, the natural log.Logarithms can be rewritten as exponential functions and exponential functions can be
rewritten as Logarithms.
logbaseanswer = exponent
baseexponent = answer
translates to
How to evaluate logarithms
Rewrite the log as an exponential function.
log5 25 5x = 25 x = 2
Some special rules for logarithms.
If the base and the answer are the same, then the exponent is 1.
log3 3 =1 ln e = 1
If the base is raised to an exponent of log with the same base then the answer of the log is the
solution.
6log6
x = x eln 4 = 4
No matter what the base, if the answer is 1 the exponent is 0.
log324 1 = 0 ln 1 = 0
More special rules for logarithms.
= x
= 0 remember 250x = 1
= 3 remember 10x = 1000
= 1 remember 5x = 5
= 2 remember 2x = 4
Now it is your turn to practice…
Evaluate:
2a. log 4
b. log1000
5c. log 5
250d. log 1
7loge. 7 x
Answer to b
Answer to c
Answer to d
Answer to e
Answer to a
For more help with evaluating logarithms visit
http://themathpage.com/alg/logarithms.htm
Expanding and Condensing
Logarithms
Properties of LogarithmsProduct Property
log log logb b buv u v
Quotient Property
log log logb b b
uu v
v
Power Property
log lognb bu n u
To Expand we use the properties of logarithms in the following order
division
multiplication
powers23
logy
xFirst we apply the quotient property
2log3 logy x Now we apply the product property2log3 log logy x Last we apply the power
property
log3 2log logy x You have successfully expanded the logarithm.
Your turn to try
Expand these logarithms
a. ln 3xy
3b. log
2
x
2c. log 5x Answer to c
ln 3 ln lnx y Answer to a
log3 log log 2x Answer to b
2log 5x
To Condense we use the properties of logarithms in the following order
powers
division
multiplication
log3 2log logy x First we apply the power property
2log3 log logy x Now we apply the quotient property2
log3 logy
x Next we apply the product
property2
log3y
xYou have successfully
condensed you logarithm
Your turn to try
Condense these logarithms
a. log3 log 6
b. log 5 + log x
c. 3log log 7x Answer to c
3 1log log
6 2 Answer to
a
log5x Answer to b
3log 7x
For more help with evaluating logarithms visit
http://themathpage.com/alg/logarithms.htm#laws
Change of Base
Up until now, the only logarithms we can use a calculator to calculate are base 10, the common log, and base e, the natural
log.
The Change of Base Formula
log lnlog or log
log lnb b
u uu u
b b
With this formula we can use a calculator no matter what the original base of the log is.
Change of BaseExample
3log 7 Apply the change of base formula
log 7
log3
ln 7
ln 3or
0.8451.771
0.477 1.946
1.7711.099
or
log15 1.1761.392
log 7 .845
log12 1.0791.544
log5 .699
Your turn to try
Apply the change of base and approximate the answer to 3 decimal
places
5a. log 12
7b. log 15
Answer to a.
Answer to b.
Finding the inverse functions of logarithms
This process is the same as finding the inverse of any function:
Step 1. Swap x and y
Step 2. Solve for y.
Example: find the inverse
5logy x Swap x and y
5logx y Solve for y by rewriting in exponential form
5xy Congratulations you have found the inverse function
5
5
3
3 log
3 log
5 x
x y
x y
y
Your turn to try
Find the inverse of the following functions
3a. logy x
5b. 3 logy x Hint: when solving for y move the 3 first.
Answer to b.
3log
3xx y
y
Answer to a.
Exponential Growth and Decay
The function represents Exponential Growth if
a > 0 and b > 1.
y = abx
The function represents Exponential Decay if
a > 0 and 0 < b < 1. Examples of
Exponential Growth
5
4 2
x
x
y
y
Examples of
Exponential Decay
13
4
8 7
x
x
y
y
Exponential Growth and Decayy = aex
When working with e, the rules for exponential growth and decay change slightly. We can now just look to the exponent to determine growth or
decay.The function represents Exponential Growth if
a > 0 and the exponent is positive.
The function represents Exponential Decay if
a > 0 and exponent is negative.
5
1
7
7
x
x
y e
y e
35
1
4
x
x
y e
y e
This exponential decay because the e has a negative exponent.
This exponential growth because the e has a positive exponent.
This exponential decay because the ½ is between 0 and 1.
Your turn to try
Determine if the function is exponential growth or decay and explain why?
21
a. 32
x
y
2b. 3 xy e
2c. 3
xy e
Answer to b.
Answer to a.
Answer to c.
Solving Exponential and Logarithmic Equations
Can we rewrite the log as an exponential function to solve for x?
2log 3x 32 x 8x
3log 2 2x 23 2x 9 2
7
x
x
Solving Exponential and Logarithmic Equations
Do we have the same base or can we rewrite the bases as common bases? If so then the exponents are equal.
3 2 510 10x x 3 2 5x x 2
2
x
x
6 29 27x x 6 22 33 3x x
2 12 63 3x x 2 12 6x x 12 4
3
x
x
Solving Exponential and Logarithmic Equations
We can NOT rewrite both sides with the same base.
Now we must take the log of both sides using the base that has the exponent.
3 14x 3 3log 3 log 14x 3log 14x Now use the change of base to give an approximation for
x.
3log 14x log14 1.1462.403
log3 0.477x
Your turn to try
Solve these equations
3 5 3a. 10 10x x
4b. log 2x
3c. log 2 1 3x
2 1d. 4 8x x
e. 2 7x
Answer to b
Answer to c
Answer to d
Answer to e
Answer to a
Answer for a
3 5 3a. 10 10x x The bases are the same so set the exponents equal to each other.
3 5 3x x Now solve for x.
2 5 3x 2 8x
4x
Answer for b
Rewrite in exponential form.
24 x Simplify
16x
4b. log 2x
Answer for c
Rewrite in exponential form.
33 2 1x Now solve for x.
9 2 1x
10 2x
3c. log 2 1 3x
5x
Answer for d
First we must rewrite the bases as the same base raised to
exponent.
2 12 32 2x x Use your rules of exponents to
simplify4 3 32 2x x
4 3 3x x
3x
2 1d. 4 8x x
Bases are the same so the exponents are equal.
Solve for x.
Answer for e
We can NOT rewrite both sides to have the same base, so we
must take the same log of both sides.
2 2log 2 log 7x Now use the properties of Logarithms to simplify the left
side
2log 7x
log 7 0.8452.807
log 2 0.301x
e. 2 7x
Use the change of base formula to approximate x.
Congratulations you have reached
the end of the review.
Good Luck on the Chapter 8 test.
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