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

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

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

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

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

6 Example 3 Express as a single logarithm. Simplify, if possible. log5625 + log525 log5 (625 • 25) 6

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

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

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

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

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

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

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

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