1. Objectives Use properties to simplify logarithmic expressions.
Translate between logarithms in any base.

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

3. Example 1: Recognizing Inverses
Simplify each expression.
a. log3311
b. log381
c. 5log510
5log510
log334
log3311 4 10 11

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

5. Example 2: Adding Logarithms
Express log64 + log69 as a single logarithm. Simplify.
log64 + log69
log6 (4  9) 2
log6 36
=______

6. 6 Example 3 Express as a single logarithm. Simplify, if possible.
log log525
log5 (625 • 25) 6

7. –1 Example 4 Express as a single logarithm. Simplify, if possible.
log log 1 3 9 1 3
log (27 • ) 9
Or you can think of it like…. –1

8. Remember that to divide powers with the same base, you subtract exponents

9. Example 5: Subtracting Logarithms
Express log5100 – log54 as a single logarithm. Simplify, if possible.
log5100 – log54 2

10. Example 6
Express log749 – log77 as a single logarithm. Simplify, if possible.
log749 – log77
ALSO,We know that
So…it =1 1

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

12. Simplifying Logarithms with Exponents
Express as a product. Simplify, if possible.
7. log2326 = ____ 8.
6log232
6(5)

13. Express as a product. Simplify, if possibly.
9. log104 =___
10. log5252 = _____
2log525
4log10
2(2)
4(1)

14. Express as a product. Simplify, if possibly.
11.