7-5 PROPERTIES OF LOGARITHMS Rolling them out and Wrapping them up.
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Transcript of 7-5 PROPERTIES OF LOGARITHMS Rolling them out and Wrapping them up.
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