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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

Click 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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