Logarithm Quotient Rule ▷ How to Rewrite a Product as a Difference

Table of Contents

The logarithm quotient rule is:

log_b(x/y) = log_b(x) – log_b(y)

b, x, y ∈ R⁺, b ≠ 1

R⁺ denotes the set of positive real numbers.

Logarithm Quotient Rule Proof

log_b(x/y) = log_b(xy^-1)

We apply the logarithm product rule:

log_b(xy^-1) = log_b(x) + log_by^-1)

We apply the logarithm power rule:

log_b(xy^-1) = -1 · log_by = – log_by

⇒ log_b(x/y) = log_bx – log_by

Logarithm Quotient Rule Examples

log₁₀(1/100) = log₁₀(1) – log₁₀(100) = 0 – 2 = -2

log(1000) = log(10000/10) = log(10000) – log(10) = 4 – 1 = 3

Binary Log Quotient Rule Examples

log₂(x/y) = log₂(x) – log₂(y)

log₂(1/8) = log₂(1) – log₂(8) = 0 – 3 = -3 log₂(0.0625) = log₂(1/16) = log₂(1) – log₂(16) = 0 – 4 = -4

Natural Log Quotient Rule Examples

ln(x/y) = ln(x) – ln(y)

ln(e/π) = ln(e) – ln(π) = 1 – 1.1447298858 = -0.14472988584 ln(2/3) = ln(2) – ln(3) = 0.6931471805 – 1.0986122886 = -0.4054651081

Frequently Asked Questions

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What is the quotient property of logarithms formula?

log(x/y) = log(x) – log(y)

How to prove the quotient rule of logarithms?

First apply the logarithm product rule, then make use of the logarithm power rule.

What is the quotient rule for logarithms?

The quotient rule for logarithms states that the log of a quotient equals the difference of the log of the dividend and the log of the divisor.

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Logarithm Quotient Rule