1. Exponential functions Logarithmic functions
Unit 7 Logarithms
Exponential functions
Logarithmic functions
Using properties of logarithms
Exponential and Logarithmic equations
Exponential and logarithmic models

2. 8.2 Solving exponential equations and inequalities

3. Solve for x: To solve exponential equations, get the bases equal.
One to one property!
Bases must be the same
Solve for x:

4. BONUS!!

5. Compound Interest Formulas
After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formula:
1) For n compounding per year:
Find the account balance after 20 years if $100 is placed in an account that pays 1.2% interest compounded twice a month.

6. If $350,000 is invested at a rate of 5½% per year, find the amount of the investment at the end of 10 years for the following compounding methods:
a) Quarterly b) Monthly

7. Solving exponential inequalities is similar to equations, make sure the bases are equal.
Solve:

8. 8.3 Logarithmic Functions
Write exponential functions as logarithms
Write logarithmic functions as exponential functions

9. A logarithm function is another way to write an exponential function
y is the logarithm, a is the base, x is the number

10. Natural log-base e Common log-base 10
When we use a common log with base 10, it is not necessary to indicate the base.
Use the log button on the calculator to take use base 10 log of any number.
Natural log-base e
e is an irrational number like π. e =
If we use a natural log, we indicate by writing ln instead of log and no base is needed.
Use the ln button on the calculator to take the natural log of any number.

11. Evaluate using Change of Base
How do we evaluate logarithms that are not common?
Change of base formula
Evaluate using Change of Base
log6 8
log3 12

12. Rewrite as a exponential equation:
log4 16 = 2
log3 729 = 6
log8 512 = 3
log16 8 = 3/4

13. Rewrite as a logarithm:
43 = 64
1251/3 = 5
113 = 1331
163/4 = 8

14. To find the exact value of a logarithm (or evaluate), we can change the equation to an exponential one.
Evaluate:
log3 81
log1/2 256

15. Evaluate:
log13 169

16. 8.4 Solving logarithmic equations

17. Solve:
Change to exponential form

18. log6x = log69

19. log3(x2 - 15) = log3 2x
log4(5x-4) > log43x

20. Solve Base e Equations Good to know ln ex = x 4 e-2x - 5 = 3
Add 5 to both sides
Divide 4 to both sides
Take ln of both sides
New property

21. Solve: 3 e4x - 12 = 15

22. Solve Natural Logs
Good to know: eln x = x
5 ln 6x = 8
3 ln 4x = 24

23. Continuously Compound Interest
A = Pert
Joan was born and her parents deposited $2000 into a college savings account paying 4% interest compounded continuously. What would be the balance after 15 years.

24. 8.5 Using Properties of logarithms
Rewrite logarithms with a different base
Use properties of logarithms to evaluate or rewrite logarithmic expressions
Use properties of logarithms to expand or condense logarithmic expressions
Use logarithmic functions to model real-life problems

25. Product Property
logx ab= logx a + logx b
Quotient Property
logx a/b = logx a - logx b
Power of Properties
logb Ax = xlogb A

26. Simplify:
Expand:

27. Use log4 2 = .5 to approximate log4 32

28. Use log37 = 1.77 to approximate log3 49

30. Solve

31. Real World Applications
The Ph of a substance is defined as the concentration of hydrogen ions [H+] in moles. It is given by the formula pH = log10(1/H+). Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2.

32. Write in log form:
ex = 9
e7 = x
Write in exponential form:
ln x = 2.143
ln 18 = x

33. Simplify the expression:
6 ln ln 4
2 ln ln 2 + ln 5y

34. To solve exponential equations, get the bases equal.
we can’t get the bases equal here.

35. Solve:
6x = 42

36. Key Chapter points: Exponential functions and graphs
Logarithmic functions
Properties of logarithms
Exponential and Logarithmic equations