Table of Contents
The logarithm product rule is:
logb(x · y) = logb(x) + logb(y)
b, x, y ∈ R+, b ≠ 1
R+ denotes the set of positive real numbers.
Logarithm Product Rule Proof
x = blogb(x)), y = blogb(y))
⇒ x · y = blogb(x) · blogb(y)
We make use of an · am = an+m
x · y = blogb(x) + logb(y)
We apply logb on both sides of the equation.⇒ logb(x · y) = logb(x) + logb(y)
Logarithm Product Rule Examples
log10(10000) = log10(100 · 100) log10(100) + log10(100) = 2 + 2 = 4
log(2.5) + log(4) = log(2.5 · 4) = log(10) = 1.
Binary Log Product Rule Examples
log2(x · y) = log2(x) + log2(y)
log2(2 · 4) = log2(2) + log2(4) = 1 + 2 = 3 log2(512) = log2(16 · 32) = log2(16) + log2(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 QuestionsClick on the question which is of interest to you to see the collapsible content answer.
What is the product rule of logarithms?
How do you use the product property of logarithms?
How do you prove the product rule of logarithms?
How do you write a logarithm as a sum or difference?
What happens if you multiply two logs?
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