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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 _{b}y ^{-1})

We apply the logarithm power rule:

log _{b}(xy ^{-1}) = -1 · log _{b}y = – log _{b}y

⇒ log _{b}(x/y) = log _{b}x – log _{b}y

## 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

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