7-5 PROPERTIES OF LOGARITHMSRolling them out and Wrapping them up

Definitions

1. Product Property

2. Quotient Property

3. Power Property

The above will be on the quiz!

Product Property

b, m, & n must be positive numbers and b ≠ 1 log b mn = log b m + log b n Examples:

log 4 21 = log 4 (3 · 7)

= log 4 3 + log 4 7 log 3 27 = log 3 (3 * 9)

= log 3 3 + log 3 9

= 1 + 2= 3

log 3 4x = log 3 4 + log 3 x

Quotient Rule

b, m, & n must be positive numbers and b ≠ 1

log b = log b m – log b n

Examples:

log 4 = log 4 3 – log 4 7

log 3 = log 3 2 – log 3 x

Notice the numerator is listed first and the

denominator is subtracted from it

mn

372x

Power Property

b, m, & n must be positive numbers and b ≠ 1

log b mn = n log b m Examples:

log 4 49 = log 4 72

= 2 log 4 7 log 2 512 = log 2 83

= 3 log 2 8

= 3 · 3= 9

Using properties to expand an expression

log 6 = log 6 5x3 – log 6 y Quotient Property

= log 6 5 + log 6 x3 – log 6 y Product Property

= log 6 5 + 3 log 6 x – log 6 y Power Property

5x3

y

Using properties to condense an expression 5 log 4 2 + 7 log 4 x – 4 log 4 y log 4 25 + log 4 x7 – log 4 y4

Power Property

log 4 25x7 – log 4 y4 Product Property

log 4 = log 4 Quotient Property & Simplify

25x7

y4

32x7

y4

Change of Base Formula

log 3 8 = ≈ ≈ 1.893

log 8log 3

0.90310.4771