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# Logarithm Product Rule

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

Click on the question which is of interest to you to see the collapsible content answer.
log(x · y) = log(x) + log(y)
The logarithm product rule states that the log of a product equals the sum of the logs of its factors.
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.
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.
The log of a any product is equal to the sum of the logs.

## Summary

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