Table of Contents

The **logarithm power rule** is:

**log _{b}(x) ^{r} = r · log _{b}(x) ** b,x ∈ R

^{+}, r ∈ R, b ≠ 1

R

^{+}denotes the set of positive real numbers.

## Logarithm Power Rule Proof

x ^{r} = (b ^{(log b(x) }) ^{r} = b ^{ r · logb (x) }

⇒ **log _{b}x ^{r} = r · log _{b}(x) **

Because, by definition of the logarithm:

b

^{y}= z ⇔ y = log

_{b}(z).

## Logarithm Power Rule Examples

log _{10}(10) ^{2} = 2 · log _{10}(10) = 2 · 1 = 2
log _{10}(10000) = log (10) ^{4} = 4 · log (10) ^{4} = 4 · 1 = 4

### Binary Log Power Rule Examples

**log _{2}(x) ^{r} = r · log _{2}(x) ** b,x ∈ R

^{+}, r ∈ R, b ≠ 1

log

_{2}(2)

^{3}= 3 · log

_{2}(2) = 3 · 1 = 3 log

_{2}(64) = log

_{2}(2)

^{6}= 6 · log

_{2}(2) = 6 · 1 = 6

### Natural Log Power Rule Examples

**ln(x) ^{r} = r · ln(x) ** b,x ∈ R

^{+}, r ∈ R, b ≠ 1

ln(e)

^{2}= 2 · ln(e) = 2 · 1 = 2 ln(25) = ln(5)

^{2}= 2 · ln(5) = 2 · 1.6094379124341005 = 3.218875824868201

## Frequently Asked Questions

Click on the question which is of interest to you to see the collapsible content answer.### How do you solve logarithmic powers?

### What is the power rule for logarithms?

_{b}(x)

^{r}= r · log

_{b}(x)

### Can logarithms be exponents?

_{b}(x) = y ⇔ b

^{ y}

### What does a number before a log mean?

_{b}(x) is equal to the exponent of log

_{b}(x)

^{r}.

### How do you convert power to log?

## Summary

If you have not already done so, please

**hit the share buttons**, and install our PWA app (see menu or sidebar).

If you still do things the traditional way: bookmark us now! BTW: Here’s the Logarithm Product Rule.

Thanks for your visit!