# Logarithms of Quotients

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

## Formula

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

## Proof

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

Log both sides. (base: *b*)

log_{b} *c* = (green) and log_{b} *d* = (brown)

Logarithmic form

*c*/*d* = *b*^{(green)}/*b*^{(brown)}

= *b*^{(green) - (brown)}

Quotient of powers

Log both sides. (base: *b*)

log_{b} *c*/*d* = (green) - (brown)

log_{b} *c* = (green) and log_{b} *d* = (brown)

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

## Example 1

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

log_{2} 2 = 1

## Example 2

The given logs' bases are the same: 6.

So log_{6} 9 - log_{6} 15 + log_{6} 10 = log_{6} (9⋅10/15).

Logarithms of products

Change the numbers to their prime factorizations

and cancel the factors.

log_{6} 6 = 1