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 Questions
Click on the question which is of interest to you to see the collapsible content answer.What is the product rule of logarithms?
log(x · y) = log(x) + log(y)
How do you use the product property of logarithms?
The logarithm product rule states that the log of a product equals the sum of the logs of its factors.
How do you prove the product rule of logarithms?
Write a x and y is exponential form as logarithms, then apply the product rule of exponents, and finally take the logarithms of both sides of the equation.
How do you write a logarithm as a sum or difference?
The logarithm product rule allows you to rewrite a log as a sum, whereas the logarithm quotient rule allows you to rewrite a log as a difference.
What happens if you multiply two logs?
The log of a any product is equal to the sum of the logs.
Summary
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