# Logarithms of Powers

How to solve logarithms of powers problems: formula, proof, examples, and their solutions.

## Basic Properties

1 = *b*^{0}.

So log_{b} 1 = 0.*b* = *b*^{1}.

So log_{b} *b* = 1.

Use these two properties to simplify logarithms.

Logarithmic Form

## Formula

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

Take the exponent *n* out from *c*^{n}.

## Proof

Set *c* = *b*^{(brown)}.

Log both sides. (base: *b*)

log_{b} *c* = (brown)

Logarithmic form

See *c* = *b*^{(brown)} again.

Do the *n*-th power on both sides.*c*^{n} = *b*^{n⋅(brown)}

Log both sides. (base: *b*)

log_{b} *c*^{n} = *n*⋅(brown)

log_{b} *c* = (brown)

So log_{b} *c*^{n} = *n*⋅log_{b} *c*.

## Example 1

Change 8 to the power of the base: 2^{3}.

Take the exponent 3 out.

log_{2} 2^{3} = 3⋅log_{2} 2

log_{2} 2 = 1

## Example 2

Change 1/81 to the power of the base: 3^{-4}.

Negative exponent

Take the exponent -4 out.

log_{3} 3^{-4} = (-4)⋅log_{3} 3

log_{3} 3 = 1

## Example 3

Change 32 to the power of the base: 2^{5}.

Take the exponent 5 out.

log_{3} 2^{5} = 5⋅log_{3} 2

log_{3} 2 = *a*

(given condition)