8.5 Properties of Logarithms

4/10/2013

Properties of Logarithms

Let m and n be positive numbers and b ≠ 1,Product Property

Quotient Property

Power Property

nmmn bbb logloglog

nmn

mbbb logloglog

mnm bn

b loglog

Expand and Condense Logarithmic Expressions

Expand: is a sum and/or difference of logs.Condense: A single log expression.

Example 2 Expand a Logarithmic Expression

Expand the expression. Assume all variables are positive.

a. log4 5x 2 log7 y3x

b.

SOLUTION

a. log4 5x 2 = log4 5 log4 x 2+ Product property

= log4 5 2 log4 x+ Power property

log7 y3x

b. = log7 3x log7 y– Quotient property

= log7 3 log7 x log7 y– Product property+

CheckpointExpand and Condense Logarithmic Expressions

Expand the expression. Assume all variables are positive.

5. log2 5x

6. log 2x 3 ANSWER log 2 + 3 log x

ANSWER log2 5 + log2 x

7. log3 7

5xANSWER log3 5 –+ log3 x log3 7

8. log6 y4x 2

ANSWER log6 4 –+ 2 log6 x log6 y

Example 3 Condense a Logarithmic Expression

Condense the expression.

a. log 16 2 log 2–

SOLUTION

a. log 16 2 log 2– = log 16 log 22– Power property

= log22

16Quotient property

= log 4 Simplify.

Example 3 Condense a Logarithmic Expression

3 log 5 log 4+ = Power propertylog 53 log 4+

Product property= ( )453 •log

Simplify. = log 500

3 log 5 log 4b. +

SOLUTION

Checkpoint

Condense the expression. Assume all variables are positive.

ANSWER log5 3

ANSWER log2 35

ANSWER log 36

9. log5 12 log5 4–

10. log2 7 log2 5+

11. log 4 2log 3+

12. 3 log x log y– ANSWER logyx 3

Expand and Condense Logarithmic Expressions

Homework:8.5 p.445 #34-56even