Home » Logarithm Product Rule

# Logarithm Product Rule

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

If you have not already done so, please press the share buttons, and install our PWA app (see menu or sidebar).

If you still do things the traditional way: bookmark us now!

Submitting...
Thank you, your sign-up request was successful! Please check your email inbox to confirm.
{{message}}
BTW: Here’s the Logarithm Quotient Rule.

Thanks for your visit!