Table of Contents
The logarithm product rule is:
log b(x · y) = log b(x) + log b(y)
b, x, y ∈ R +, b ≠ 1
R + denotes the set of positive real numbers.
Logarithm Product Rule Proof
x = b log b(x) ), y = b log b(y) )
⇒ x · y = b log b(x) · b log b(y)
We make use of a n · a m = a n+m
x · y = b log b(x) + log b(y)
We apply log b
on both sides of the equation. ⇒ log b(x · y) = log b(x) + log b(y)
Logarithm Product Rule Examples
log 10(10000) = log 10(100 · 100) log 10(100) + log 10(100) = 2 + 2 = 4
log(2.5) + log(4) = log(2.5 · 4) = log(10) = 1.
Binary Log Product Rule Examples
log 2(x · y) = log 2(x) + log 2(y)
log 2(2 · 4) = log 2(2) + log 2(4) = 1 + 2 = 3
log 2(512) = log 2(16 · 32) = log 2(16) + log 2(32) = 4 + 5 = 9
Natural Log Product Rule Examples
ln(x · y) = ln(x) + ln(y)
ln(π · e) = ln(π) + ln(e) = 1.1447298858 + 1 = 2.1447298858
ln(1/4e) + ln(4e) = ln(1/4e · 4e) = ln (1) = 0
Frequently Asked Questions
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