1. Properties of Logarithms
These properties are based on
rules of exponents
since
logs =
exponents

2. I. 𝑙𝑜𝑔 𝑏 1=0 Because in exponential form 𝑏 0 =1 𝑙𝑜𝑔 5 1= 𝑙𝑜𝑔 𝑚 1=
(any number to the zero power = 1)
5 to what power = 1?
Example:
𝑙𝑜𝑔 5 1=
Example:
𝑙𝑜𝑔 𝑚 1=

3. II. 𝑙𝑜𝑔 𝑏 𝑏=1 1 1 Because in exponential form 𝑏 1 =𝑏 𝑙𝑜𝑔 5 5= 𝑙𝑜𝑔 𝑚 𝑚=
(any number to the first power is itself)
5 to what power = 5? 1
Example:
𝑙𝑜𝑔 5 5= 1
Example:
𝑙𝑜𝑔 𝑚 𝑚=

4. III. Product Rule 𝑙𝑜𝑔 𝑏 𝑚𝑛= 𝑙𝑜𝑔 𝑏 𝑚+𝑙𝑜𝑔 𝑏 𝑛 𝑙𝑜𝑔 𝑏 𝑥𝑦 =
𝑙𝑜𝑔 𝑏 𝑚𝑛= 𝑙𝑜𝑔 𝑏 𝑚+𝑙𝑜𝑔 𝑏 𝑛
Because in exponential form 𝑏 𝑚 ×𝑏 𝑛 = 𝑏 𝑚+𝑛
Examples:
𝑙𝑜𝑔 𝑏 𝑥𝑦 =
𝑙𝑜𝑔 𝑏 𝑥+ 𝑙𝑜𝑔 𝑏 𝑦
𝑙𝑜𝑔6 =
𝑙𝑜𝑔2+𝑙𝑜𝑔3
𝑙𝑜𝑔 3 9𝑏 =
𝑙𝑜𝑔 3 9+ 𝑙𝑜𝑔 3 𝑏

5. IV. Quotient Rule 𝑙𝑜𝑔 𝑏 𝑚 𝑛 = 𝑙𝑜𝑔 𝑏 𝑚−𝑙𝑜𝑔 𝑏 𝑛 𝑙𝑜𝑔 5 𝑥 𝑦 =
𝑙𝑜𝑔 𝑏 𝑚 𝑛 = 𝑙𝑜𝑔 𝑏 𝑚−𝑙𝑜𝑔 𝑏 𝑛
Because in exponential form 𝑏 𝑚 𝑏 𝑛 = 𝑏 𝑚−𝑛
Examples:
𝑙𝑜𝑔 5 𝑥 𝑦 =
𝑙𝑜𝑔 5 𝑥− 𝑙𝑜𝑔 5 𝑦
𝑙𝑜𝑔 2 𝑎 3 =
𝑙𝑜𝑔 2 𝑎− 𝑙𝑜𝑔 2 3
𝑙𝑜𝑔 3 6𝑏 7 =
𝒍𝒐𝒈 𝟑 𝟔+ 𝒍𝒐𝒈 𝟑 𝒃− 𝒍𝒐𝒈 𝟑 𝟕

6. V. Power Rule 𝑙𝑜𝑔 𝑏 𝑚 𝑛 =𝑛 𝑙𝑜𝑔 𝑏 𝑚 3𝑙𝑜𝑔 2 𝑎+ 4𝑙𝑜𝑔 2 𝑏 𝑙𝑜𝑔 5 𝑥 3 =
𝑙𝑜𝑔 𝑏 𝑚 𝑛 =𝑛 𝑙𝑜𝑔 𝑏 𝑚
Because in exponential form 𝑏 𝑚 𝑛 = 𝑏 𝑚𝑛
Examples:
𝑙𝑜𝑔 5 𝑥 3 =
3 𝑙𝑜𝑔 5 𝑥
𝑙𝑜𝑔 2 𝑎 3 𝑏 4 =
3𝑙𝑜𝑔 2 𝑎+ 4𝑙𝑜𝑔 2 𝑏

7. 𝑙𝑜𝑔 𝑏 𝑚= 𝑙𝑜𝑔𝑚 𝑙𝑜𝑔𝑏 𝑙𝑜𝑔9 𝑙𝑜𝑔5 𝑙𝑜𝑔 5 9 = VI. Change of Base Formula
Example:
𝑙𝑜𝑔 =
These properties remain the same when working with the natural log.

8. True or False: True False True True False False False False True True
Use properties of logarithms to determine if each of the following is true or false. Check your answers using your calculator
True or False:
True
False
True
True
False
False
False
False
True
True
True
True

9. Use the properties of logs to expand the following expressions:1.
1. Apply Product Rule:
2. Apply Power Rule:

10. Use the properties of logs to expand the following expressions:2.
1. Apply Product Rule:
2. Apply Power Rule:

11. Use the properties of logs to expand the following expressions:3.
1. Apply Quotient Rule:
2. Apply Product Rule:

12. Use the properties of logs to expand the following expressions:4.
1. Change radical to exponential form:
2. Apply Product Rule:
3. Apply Power Rule:

13. Use the properties of logs to expand the following expressions:5.
2. Apply Product Rule:
3. Apply Power Rule:

14. Write as a single logarithmic expression.5.
1. Apply Reverse Power Rule:
2. Apply Reverse Quotient Rule:
3. Change to radical form

15. Write as a single logarithmic expression.6.
1. Apply Reverse Product Rule:
2. Simplify

16. Write as a single logarithmic expression.
1. Apply Reverse Power Rule: 6.
2. Apply Reverse Product Rule:

17. Practice Time