Logarithms of Quotients

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

logb c/d = logb c - logb d

Proof

Logarithms of Quotients: 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

c/d = b(green)/b(brown)
= b(green) - (brown)

Quotient of powers

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

logb c = (green) and logb d = (brown)
So logb c/d = logb c - logb d.

Example 1

Simplify the given expression. log_2 (32/sqrt(8))

log2 (32/√8) = log2 32 - log28

Logarithms of powers

log2 2 = 1

Example 2

Simplify the given expression. log_6 9 - log_6 15 + log_6 10

The given logs' bases are the same: 6.

So log6 9 - log6 15 + log6 10 = log6 (9⋅10/15).

Logarithms of products

Change the numbers to their prime factorizations
and cancel the factors.

log6 6 = 1