6-2: Properties of Logarithms Unit 6: Exponents/Logarithms English Casbarro

You can find these properties because of exponential rules. a. b m b n = b m+n b. b m = b m-n b n c. (b m ) n = b mn

Example 1: Express as a single logarithm. Simplify. a. log log 4 32b. log log 5 25 c.

Example 2: Express as a single logarithm. Simplify. a. log 2 32 – log 2 4b. log 7 49 – log 7 7 c. log 2 16 – log 2 2

Example 3: Express as a single logarithm. Simplify. a. log b. c. log

Exponential and Logarithmic are inverses, so They “undo” each other. AlgebraExample log b b x = x log = 7 b log b x = x 10 log 10 2 = 2 For any base b, such that b > 0 and b ≠ 1

Change of base formula For a > 0 and a ≠ 1 and any base b, such that b > 0 and b ≠ 1 AlgebraExample Note: This is most often used to change the base to 10 or e so that you can use your calculator.

Example 4: Simplify each expression. a. b. c. d. e. f.

Not all logarithms involve strictly numbers; some also involve variables. The properties work exactly the same way. Example 5: Write as a single logarithm.

These properties are used to evaluate expressions as well. Example 6: a. b. c. d.

Example 7: Solve the following problem for x using the properties.

Example 8: Solve the following problem for x by using the properties.