# Logarithm of a Product

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

## Formula

log_{b} *c*⋅*d* = log_{b} *c* + log_{b} *d*

## Example 1: log_{2} 3 = *a*, log_{2} 24 = ?

Change 24 to its prime factorization.

Then 24 = 2^{3}⋅3.

prime factorization

Split the log into two parts.

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

log_{2} 2^{3} = 3⋅log_{2} 2

Logarithm of a power

log_{2} 2 = 1

Logarithm of the base

log_{2} 3 = *a*

So (given) = 3⋅1 + *a*.

Arrange the terms.

Then (given) = *a* + 3.

## Example 2: Simplify log_{6} 4 + log_{6} 9

The bases of both logs are the same: 6.

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

4⋅9 = 36

36 = 6^{2}

log_{6} 6^{2} = 2⋅log_{6} 6

Logarithm of a power

log_{6} 6 = 1

Logarithm of the base

2⋅1 = 2

So (given) = 2.