Quotient Rule of Logarithms
Simple Explanation
The logarithm of a quotient equals the difference of the logarithms: log_b(M/N) = log_b M − log_b N.
Why Do We Need It?
The quotient rule is the division counterpart to the product rule, letting you split or combine logarithms of fractions the same way you split or combine logarithms of products.
Formula
Quotient Rule of Logarithms
log_b(M/N) = log_b M − log_b N
The logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.
- b
- — the base of the logarithm (b > 0, b ≠ 1)
- M, N
- — positive real numbers, with M divided by N
When to use it: Whenever you need to expand the logarithm of a quotient into a difference, or condense a difference of logarithms into one logarithm of a quotient.
Worked Example
Expand a logarithm of a quotient
Expand log₅(125/25) using the quotient rule, then evaluate.
Why Does This Work?
If log_b M = p and log_b N = q, then M = bᵖ and N = bᵠ, so M/N = bᵖ/bᵠ = bᵖ⁻ᵠ by the exponent quotient rule — meaning log_b(M/N) is exactly p − q.
Real-Life Example
Comparing pH readings
pH is a base-10 logarithmic scale, and comparing two solutions often involves the ratio of their hydrogen-ion concentrations.
log₁₀ of that concentration ratio equals the difference of the two pH-defining logarithms, matching the quotient rule directly.
Practice
Which expression equals log₂(x/8)?
MediumCommon mistake
Subtracting in the wrong order — log_b(M/N) is log_b M minus log_b N, not the other way around; the numerator's logarithm always comes first.
Quick Review
- log_b(M/N) = log_b M − log_b N.
- The quotient rule converts division inside a log into subtraction outside it.
- Keep the order: numerator's log minus denominator's log.