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_{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

## 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?

### How to prove the quotient rule of logarithms?

### What is the quotient rule for logarithms?

## Summary

If you have not already done so, please

**hit the share buttons**, and install our PWA app (see sidebar or menu).

If you still do things the traditional way: bookmark us now!BTW: Here’s the Change of Base Law.

Thanks for your visit!