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# Logarithm Quotient Rule

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

log 10(1/100) = log 10(1) – log 10(100) = 0 – 2 = -2

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

### Binary Log Quotient Rule Examples

log 2(x/y) = log 2(x) – log 2(y)

log 2(1/8) = log 2(1) – log 2(8) = 0 – 3 = -3 log 2(0.0625) = log 2(1/16) = log 2(1) – log 2(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

Click on the question which is of interest to you to see the collapsible content answer.
log(x/y) = log(x) – log(y)
First apply the logarithm product rule, then make use of the logarithm power rule.
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.

## Summary

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