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# Logarithm Quotient Rule

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

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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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BTW: Here’s the Change of Base Law.

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