Download Log Properties

7-5 PROPERTIES OF
LOGARITHMS
Rolling them out and Wrapping them up
Definitions

1. Product Property
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
2. Quotient Property


3. Power Property
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
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 m = log b m – log b n
n
Examples:

3 = log 4 3 – log 4 7
7
 log 3
2 = log 3 2 – log 3 x
x
Notice the numerator is listed first and the

 log 4
denominator is subtracted from it
Power Property



b, m, & n must be positive numbers and b ≠ 1
log b mn = n log b m
Examples:
49 = log 4 72
= 2 log 4 7
 log 2 512 = log 2 83
= 3 log 2 8
=3·3
=9
 log 4
Using properties to expand an expression

log 6
5x3 = log 6 5x3 – log 6 y
Quotient Property
y
= log 6 5 + log 6 x3 – log 6 y Product Property
= log 6 5 + 3 log 6 x – log 6 y Power Property
Using properties to condense an expression
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
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
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
7
5
7
32x
log 4 24 x = log 4 4
Quotient Property &
y
y
Simplify
Change of Base Formula
 log 3 8 = log 8 ≈ 0.9031
log 3
0.4771
≈ 1.893