# Logarithm of a Power

How to solve the logarithm of a power (log_{a} x^{m}): formula, 3 examples, and their solutions.

## Log of 1 (log_{a} 1)

### Formula

a^{0} = 1

Zero Exponent

The exponent is 0.

So log_{a} 1 = 0.

The logarithm of 1 is 0.

Logarthmic Form

This formula is used

when solving logarithm problems.

## Log of Itself (log_{a} a)

### Formula

a^{1} = a

The exponent is 1.

So log_{a} a = 1.

The logarithm of itself is 1.

This formula is also used

when solving logarithm problems.

## Formula

### Formula

log_{a} x^{m} = m log_{a} x

Take the exponent m

out from the log.

## Example 1

### Example

### Solution

8 = 2^{3}

Power

Take the exponent 3

out from the log.

log_{2} 2^{3} = 3 log_{2} 2

log_{2} 2 = 1

3⋅1 = 3

So 3 is the answer.

This means 2^{3} = 8.

## Example 2

### Example

### Solution

1/81 = 1/3^{4}

1/3^{4} = 3^{-4}

Negative Exponent

Take the exponent -4

out from the log.

log_{3} 3^{-4} = -4 log_{3} 3

log_{3} 3 = 1

-4⋅1 = -4

So -4 is the answer.

This means 3^{-4} = 1/81.

## Example 3

### Example

### Solution

32 = 2^{5}

Take the exponent 5

out from the log.

log_{3} 2^{5} = 5 log_{3} 2

log_{3} 2 = a

So 5 log_{3} 2 = 5⋅a.

5⋅a = 5a

So 5a is the answer.