# Logarithm of a Power

How to solve the logarithm of a power: formula, examples, and their solutions.

## Logarithm of 1

log_{b} 1 = 0

This is true because

1 = *b*^{0}.

This formula is used

to simplify logarthims.

Logarithmic Form

## Logarithm of the Base

log_{b} *b* = 0

This is true because*b* = *b*^{1}.

This formula is also used

to simplify logarthims.

## Formula

log_{b} *c*^{n} = *n* log_{b} *c*

To solve the logarithm of a power,

take the exponent *n*

out from the log.

## Example 1: Simplify log_{2} 8

Change 8 to the power of the base:

2^{3}.

Take the exponent 3

out from the log.

Then (given) = 3⋅log_{2} 2.

The logarithm of the base is 1.

So log_{2} 2 = 1.

So (given) = 3⋅1.

3⋅1 = 3

So (given) = 3.

## Example 2: Simplify log_{3} (1/81)

Change 1/81 to the power of the base:

1/3^{4}.

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

Negative exponent

Take the exponent -4

out from the log.

Then (given) = -4⋅log_{3} 3.

The logarithm of the base is 1.

So log_{3} 3 = 1.

So (given) = -4⋅1.

-4⋅1 = -4

So (given) = -4.

## Example 3: log_{3} 2 = *a*, log_{3} 32 = ?

log_{3} 2 = *a*

The base of the given log is 2.

So change 32 to the power of 2:

2^{5}.

Take the exponent 5

out from the log.

Then (given) = 5⋅log_{3} 2.

log_{3} 2 = *a*

So (given) = 5⋅*a*.

So (given) = 5*a*.