Logarithms of Products

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

logb cd = logb c + logb d

Proof

Logarithms of Products: Proof of the formula

Set c = b(green) and d = b(brown).

Log both sides. (base: b)
logb c = (green) and logb d = (brown)

Logarithmic form

cd = b(green)b(brown)
= b(green) + (brown)

Product of powers

Log both sides. (base: b)
logb cd = (green) + (brown)

logb c = (green) and logb d = (brown)
So logb cd = logb c + logb d.

Example 1

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

Change 24 to its prime factorization: 23⋅3.

log2 23⋅3 = log2 23 + log2 3

Logarithms of powers

log2 2 = 1

log2 3 = a
(given condition)

Example 2

Simplify the given expression. log_6 4 + log_6 9

Both logs' bases are the same: 6.

So log6 4 + log6 9 = log6 4⋅9.

Logarithms of powers

log6 6 = 1