1. Warm - up

2. Properties of Logarithms
Chapter 3 Sec 3
Properties of Logarithms

3. Change-of-base formula
Essential Question
How do you rewrite logarithmic expressions to simplify or evaluate them?
Key Vocabulary:
Change-of-base formula

4. Change of Base Formula
This allows you to write equivalent logarithmic expressions that have different bases. For example change base 3 into base 10

5. Change of Base
Example 1
Change bases using common logarithms. Then approximate its value.
Change bases using natural logarithms. Then approximate its value.

6. Properties of Logarithms
Let a be a positive number such that a ≠ 1, and let n be a real number. If u and v are positive real numbers, the following properties are true.
Product Property:
Quotient Property:
Power Property:

7. Example 2 Write each logarithm in terms of ln 2 and ln 3.
Use properties of logarithms to verify:

8. Example 3
Expand each logarithmic expression.

9. Example 4
Condense each logarithmic expression.

10. Essential Question
How do you rewrite logarithmic expressions to simplify or evaluate them?

11. Solving Exponential and Logarithmic Equations
Chapter 3 Sec 4
Solving Exponential and Logarithmic Equations
Essential Question
How do you solve exponential and logarithmic equations?

12. Solving Equations One-to-One ax = ay if and only if x = y
loga x = loga y if and only if x = y
Inverse Properties

13. 2x = 32 ln x – ln 3 = 0 ex = 7 ln x = –3 log x = –1 2x = 25
Example 1
Original
Equation
Rewritten
Equation
Solution
Property
2x = 32
ln x – ln 3 = 0
ex = 7
ln x = –3
log x = –1
2x = 25
ln x = ln 3
3–x = 32
ln ex = ln 7
eln x = e–3
10log x = 10–1
x = 5
x = 3
x = –2
x = 7
x = e–3
x = 10–1 = 0.1
One-to-One
Inverse

14. Solve each equation. a. ex = 72 b. 3(2x) = 42 ln ex = ln 72 2x = 14
Example 2
a. ex = 72
ln ex = ln 72
x = ln 72 ~ 4.28
b. 3(2x) = 42
2x = 14
log22x = log214
x = log214

15. Solve the equation.
Example 3

16. Solve the equation using quadratics.
Example 4

17. Solve each equation.
Example 5
a. ln 3x = 2
eln 3x = e2
3x = e2

18. Solve the equation check for extraneous solutions.
Example 6
Solve the equation check for extraneous solutions.
Check: X
ln (–1) is invalid.

19. Solve each equation.
Example 5
You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?

20. How do you solve exponential and logarithmic equations?
Essential Question
How do you solve exponential and logarithmic equations?

21. Chapter 3.3 -3.4 Text Book Daily Assignment Pgs 211 – 212
#1 – 45 and 59 – 73 Mode 4 (1,5,9,13…73)
Pgs 221 – 222
#1 – 21 Mode 4; #29 – 49 Mode 4; #85 – 97 Mode 4
Read Section 3.5

22. Ch 3a Pop Quiz