# Logarithm of a Quotient

How to solve the logarithm of a quotient (log_{a} x/y): formula, 2 examples, and their solutions.

## Formula

log_{a} x/y = log_{a} x - log_{a} y

You can split the log of a quotient like this.

## Examplelog_{2} 32/√8

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

32 = 2^{5}

√8 = √2^{3}

Power

√2^{3} = 2^{3/2}

Rational Exponent

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

Logarithm of Itself

5⋅1 = 5

-[3/2]⋅1 = -3/2

5 = 10/2

10/2 - 3/2 = 7/2

So 7/2 is the answer.

## Examplelog_{6} 9 - log_{6} 15 + log_{6} 10

Every term has log_{6}.

So combine these logs into log_{6}.

First write log_{6} (.

log_{6} 9

The sign is plus.

So write 9.

-log_{6} 15

The sign is minus.

So divide 15.

+log_{6} 10

The sign is plus.

So multiply 10.

Logarithm of a Product

So

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

= log_{6} ([9/15]⋅10).

Reduce 10 to, 10/5, 2

and reduce 15 to, 15/5, 3.

9/3 = 3

3⋅2 = 6

log_{6} 6 = 1

So 1 is the answer.