1. Logarithmic and Exponential Functions
By: Heather McGuire
3/21/08

2. Cobb County Algebra 2 Standards
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.

3. The number e
Review
The rules of exponents we have learned in the last 2 chapters also apply to e. Click the sound buttons to here the descriptions.

4. Answer for a. Answer for b. Answer for c.
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.
Answer for a.
Answer for b.
Answer for c.

5. Answer for a First you multiply the numbers.
Then add the exponents of e.
Congratulations you did it.
Click the u-turn arrow to return to the problem page.

6. Answer for b First you divide the numbers.
Then subtract the exponents of e and place e where the largest exponent was.
Congratulations you did it.
Click the u-turn arrow to return to the problem page.

7. 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.
Click the u-turn arrow to return to the problem page.

8. Logarithms
Review
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
translates to
baseexponent = answer

9. How to evaluate logarithms
Rewrite the log as an exponential function.
log  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 = ln e = 1

10. More special rules for logarithms.
No matter what the base, if the answer is 1 the exponent is 0.
log324 1 = ln 1 = 0
If the base is raised to an exponent of log with the same base then the answer of the log is the solution.
6log6x = x eln 4 = 4

11. Now it is your turn to practice…
Evaluate:
= 2 remember 2x = 4
Answer to a
= 3 remember 10x = 1000
Answer to b
= 1 remember 5x = 5
Answer to c
= 0 remember 250x = 1
Answer to d
= x
Answer to e

12. For more help with evaluating logarithms visit

13. Expanding and Condensing Logarithms
Properties of Logarithms
Product Property
Quotient Property
Power Property

14. To Expand we use the properties of logarithms in the following order
division
multiplication
powers
First we apply the quotient property
Now we apply the product property
Last we apply the power property
You have successfully expanded the logarithm.

15. Expand these logarithms
Your turn to try
Expand these logarithms
Answer to a
Answer to b
Answer to c

16. To Condense we use the properties of logarithms in the following order
powers
division
multiplication
First we apply the power property
Now we apply the quotient property
Next we apply the product property
You have successfully condensed you logarithm

17. Condense these logarithms
Your turn to try
Condense these logarithms
Answer to a
Answer to b
Answer to c

18. For more help with evaluating logarithms visit

19. The Change of Base Formula
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
With this formula we can use a calculator no matter what the original base of the log is.

20. Change of Base
Example
Apply the change of base formula or or

21. Your turn to try
Apply the change of base and approximate the answer to 3 decimal places
Answer to a.
Answer to b.

22. 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
Swap x and y
Solve for y by rewriting in exponential form
Congratulations you have found the inverse function

23. Find the inverse of the following functions
Your turn to try
Find the inverse of the following functions
Answer to a.
Answer to b.
Hint: when solving for y move the 3 first.

24. Exponential Growth and Decay
y = abx
The function represents Exponential Growth if
a > 0 and b > 1.
The function represents Exponential Decay if
a > 0 and 0 < b < 1.
Examples of
Exponential Growth
Examples of
Exponential Decay

25. Exponential Growth and Decay
y = 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.

26. Your turn to try
Determine if the function is exponential growth or decay and explain why?
This exponential decay because the ½ is between 0 and 1.
Answer to a.
This exponential growth because the e has a positive exponent.
Answer to b.
This exponential decay because the e has a negative exponent.
Answer to c.

27. Solving Exponential and Logarithmic Equations
Can we rewrite the log as an exponential function to solve for x?

28. 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.

29. 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.
Now use the change of base to give an approximation for x.

30. Your turn to try
Solve these equations
Answer to a
Answer to b
Answer to c
Answer to d
Answer to e

31. Answer for a
The bases are the same so set the exponents equal to each other.
Now solve for x.

32. Answer for b
Rewrite in exponential form.
Simplify

33. Rewrite in exponential form.
Answer for c
Rewrite in exponential form.
Now solve for x.

34. First we must rewrite the bases as the same base raised to exponent.
Answer for d
First we must rewrite the bases as the same base raised to exponent.
Use your rules of exponents to simplify
Bases are the same so the exponents are equal.
Solve for x.

35. Answer for e
We can NOT rewrite both sides to have the same base, so we must take the same log of both sides.
Now use the properties of Logarithms to simplify the left side
Use the change of base formula to approximate x.

36. Congratulations you have reached the end of the review.
Good Luck on the Chapter 8 test.