1. How do we use properties to simplify logarithmic expressions?
Properties of Logarithms
Essential Questions
How do we use properties to simplify logarithmic expressions?
How do we translate between logarithms in any base?
Holt McDougal Algebra 2
Holt Algebra 2

2. The logarithmic function for pH that you saw in the previous lessons, pH =–log[H+], can also be expressed in exponential form, as 10–pH = [H+].
Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents

3. Remember that to multiply powers with the same base, you add exponents.

4. The property in the previous slide can be used in reverse to write a sum of logarithms (exponents) as a single logarithm, which can often be simplified.
Helpful Hint
Think: log j + log a + log m = log jam

5. log64 + log69 Express as a single logarithm. Simplify, if possible.
Adding Logarithms
Express as a single logarithm. Simplify, if possible.
log64 + log69
To add the logarithms, multiply the numbers.
log6 (4  9)
log6 36
= x
Simplify. 2
Think: 6 x = 36.

6. Adding Logarithms
Express as a single logarithm. Simplify, if possible.
log log525
To add the logarithms, multiply the numbers.
log5 (625 • 25)
log5 15,625
Simplify.
= x 6
Think: 5 x =

7. Express as a single logarithm. Simplify, if possible.
Adding Logarithms
Express as a single logarithm. Simplify, if possible.
To add the logarithms, multiply the numbers.
= x
Simplify.
Think: x = 3 1 3 –1

8. Remember that to divide powers with the same base, you subtract exponents
Because logarithms are exponents, subtracting logarithms with the same base is the same as finding the logarithms of the quotient with that base.

9. The property above can also be used in reverse.
Just as a5b3 cannot be simplified, logarithms must have the same base to be simplified.
Caution

10. Subtracting Logarithms
Express as a single logarithm. Simplify, if possible.
To subtract the logarithms, divide the numbers.
= x
Simplify. 2
Think: 5 x = 25.

11. Subtracting Logarithms
Express as a single logarithm. Simplify, if possible.
To subtract the logarithms, divide the numbers.
= x
Simplify. 1
Think: 7 x = 7.

12. Because you can multiply logarithms, you can also take powers of logarithms.

13. Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
Powers expressed as multiplication.
Think: 2 x = 32. 30
Simplify.

14. Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
Powers expressed as multiplication.
Think: 16 x = 4. 10
Simplify.

15. Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
Powers expressed as multiplication.
Think: 10 x = 10. 4
Simplify.

16. Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
Powers expressed as multiplication.
Think: 5 x = 25. 4
Simplify.

17. Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
Powers expressed as multiplication.
Think: 2 x = ½ . -5
Simplify.

18. Exponential and logarithmic operations undo each other since they are inverse operations.

19. Recognizing Inverses
Simplify each expression.
12. log381
13. 5log510
11. log3311
log3311
log334
5log510 11 4 10
14. log100.9
15. 2log2(8x)
Log
2log2(8x)
0.9 8x

20. Most calculators calculate logarithms only in base 10 or base e
Most calculators calculate logarithms only in base 10 or base e. You can change a logarithm in one base to a logarithm in another base with the following formula.

21. Changing the Base of a Logarithm
16. Evaluate log328.
17. Evaluate log927.
Change to base 10
Change to base 10
log328 =
log8
log32
log927 =
log27
log9
≈ 0.6
≈ 1.5
18. Evaluate log816.
Change to base 10
log816 =
log16
log8
≈ 1.3

22. Lesson 9.1 Practice A