Logarithm of a Quotient

Logarithm of a Quotient

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

Formula

log_b (c/d) = log_b c - log_b d

logb c/d = logb c - logb d

Example 1: Simplify log2 32/√8

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

Split the log into two parts.

log2 32/√8 = log2 32 - log28

32 = 25

8 = √23

23 = 23/2

Rational exponents

log2 25 = 5 log2 2

log2 23/2 = (3/2) log2 2

Logarithm of a power

log2 2 = 1

So (given) = 5⋅1 - (3/2)⋅1.

Logarithm of the base

5⋅1 = 5 = 10/2

10/2 - 3/2 = 7/2

So (given) = 7/2.

Example 2: Simplify log6 9 - log6 15 + log6 10

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

The bases of the logs are the same: 6.

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

Logarithm of a product

To simplify this fraction,
write the prime factorization of 9, 10, and 15.

9 = 3⋅3
10 = 2⋅5
15 = 3⋅5

Prime factorization

Cancel the factors 3 and 5
that are in both of the numerator and the denominator.

Then (given) = log6 (3⋅2).

3⋅2 = 6

log6 6 = 1

So (given) = 1.

Logarithm of the base