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
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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