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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^{(logb(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?

The value of any log with a given argument in exponential form is equal to the exponent multiplied by the logarithm of the argument.

### What is the power rule for logarithms?

log

_{b}(x)^{r}= r · log_{b}(x)### Can logarithms be exponents?

Yes, because y = log

_{b}(x) = y ⇔ b^{ y}### What does a number before a log mean?

The coefficient r in r · log

_{b}(x) is equal to the exponent of log_{b}(x)^{r}.### How do you convert power to log?

The inverse of a logarithmic function is an exponential function – and vice versa. In other words, the two functions undo each other.

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