Logarithm of a Product

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

logb cd = logb c + logb d

Example 1: log2 3 = a, log2 24 = ?

If log_2 3 = a, find the value of the given expresssion. log_2 24

Change 24 to its prime factorization.

Then 24 = 23⋅3.

prime factorization

Split the log into two parts.

log2 23⋅3 = log2 23 + log2 3

log2 23 = 3⋅log2 2

Logarithm of a power

log2 2 = 1

Logarithm of the base

log2 3 = a

So (given) = 3⋅1 + a.

Arrange the terms.

Then (given) = a + 3.

Example 2: Simplify log6 4 + log6 9

Simplify the given expression. log_6 4 + log_6 9

The bases of both logs are the same: 6.

So log6 4 + log6 9 = log6 4⋅9.

4⋅9 = 36

36 = 62

log6 62 = 2⋅log6 6

Logarithm of a power

log6 6 = 1

Logarithm of the base

2⋅1 = 2

So (given) = 2.