# Logarithms of Products

How to solve logarithms of products 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)}

Product 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

Change 24 to its prime factorization: 2^{3}⋅3.

log_{2} 2^{3}⋅3 = log_{2} 2^{3} + log_{2} 3

log_{2} 2 = 1

log_{2} 3 = *a*

(given condition)

## Example 2

Both logs' bases are the same: 6.

So log_{6} 4 + log_{6} 9 = log_{6} 4⋅9.

log_{6} 6 = 1