6-2: Properties of 6-2: Properties of LogarithmsLogarithms

Unit 6: Exponents/LogarithmsUnit 6: Exponents/LogarithmsEnglish CasbarroEnglish Casbarro

You can find these properties because of exponential rules.

a. bmbn= bm+n

b. bm= bm-n

bn

c. (bm)n= bmn

Example 1: Express as a single logarithm.

Simplify.a. log42 + log432 b. log5625 + log525 c. 1 1

3 3

127 +

9log log

Example 2: Express as a single logarithm. Simplify.

a. log232 – log24 b. log749 – log77 c. log216 – log22

Example 3: Express as a single logarithm. Simplify.

a. log3814 b. c. log5252315 5log ( )

Exponential and Logarithmic are inverses, soThey “undo” each other.

AlgebraAlgebra ExampleExampleloglogbbbbxx = x = x loglog1010101077 = 7 = 7bbloglogbbxx= x= x 1010loglog101022= 2= 2

For any base b, such that b > 0 and b ≠ 1

Change of base Change of base formulaformula

For a > 0 and a ≠ 1 and any base b, such that b > 0 and b ≠ 1

AlgebraAlgebra ExampleExample

=loglog log

ab

a

xx

b2

42

88 =

4loglog log

Note: This is most often used to change the base to 10 or e so thatyou can use your calculator.

Example 4: Simplify each expression.

a. b. c. 3 +18 8log x

5 125log 2722log

d. e. f. 4 8log 9 27log8 16log

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.

23 3+ 7 - 5 = 6 +1log ( ) log ( )x x x

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

2 3 + - 4 = + 8log log ( ) log ( )a a ax x