Table of Contents
The logarithm quotient rule is:
logb(x/y) = logb(x) – logb(y)
b, x, y ∈ R+, b ≠ 1
R+ denotes the set of positive real numbers.
Logarithm Quotient Rule Proof
logb(x/y) = logb(xy-1)
We apply the logarithm product rule:
logb(xy-1) = logb(x) + logby-1)
We apply the logarithm power rule:
logb(xy-1) = -1 · logby = – logby
⇒ logb(x/y) = logbx – logby
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
Click on the question which is of interest to you to see the collapsible content answer.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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