# Logarithm of a Quotient

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

## Formula

log_{b} *c*/*d* = log_{b} *c* - log_{b} *d*

## Example 1: Simplify log_{2} 32/√8

Split the log into two parts.

log_{2} 32/√8 = log_{2} 32 - log_{2} √8

32 = 2^{5}

√8 = √2^{3}

√2^{3} = 2^{3/2}

Rational exponents

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

log_{2} 2^{3/2} = (3/2) log_{2} 2

Logarithm of a power

log_{2} 2 = 1

So (given) = 5⋅1 - (3/2)⋅1.

Logarithm of the base

5⋅1 = 5 = 10/2

10/2 - 3/2 = 7/2

So (given) = 7/2.

## Example 2: Simplify log_{6} 9 - log_{6} 15 + log_{6} 10

The bases of the logs are the same: 6.

So log_{6} 9 - log_{6} 15 + log_{6} 10

= log_{6} 9⋅10/15.

Logarithm of a product

To simplify this fraction,

write the prime factorization of 9, 10, and 15.

9 = 3⋅3

10 = 2⋅5

15 = 3⋅5

Prime factorization

Cancel the factors 3 and 5

that are in both of the numerator and the denominator.

Then (given) = log_{6} (3⋅2).

3⋅2 = 6

log_{6} 6 = 1

So (given) = 1.

Logarithm of the base