Logarithmic Quotient Rule

July 4, 2026

A quick overview of the logarithmic quotient rule.

Quotient Rule For Logarithms

The logarithmic quotient rule is

\[\log_a\left(\frac{x}{y}\right)=\log_a(x)-\log_a(y)\] \[\text{Assuming: }x>0,\qquad y>0,\qquad a>0,\qquad a\ne 1\]

When you have division within a logarithm, you can rewrite it as the difference between two logarithms of the same base where you subtract the logarithm that contains the denominator as its argument from the logarithm that has the numerator as its argument.

Watch this section: quotient rule for logarithms at 0:00.

Intuition Behind Rule (This is a combination of two other rules in one!)

Start with

\[\log_a\left(\frac{x}{y}\right)\]

We can rewrite the fraction inside the logarithm as

\[\frac{x}{y}=x\left(\frac{1}{y}\right)\]

Then we can rewrite $\frac{1}{y}$ using the negative exponent rule

\[\frac{1}{y}=y^{-1}\]

\[\log_a\left(\frac{x}{y}\right)=\log_a(xy^{-1})\]

Using this new form, lets expand using the logarithmic product rule (Product Rule: $\log_a(xy)\rightarrow \log_a(x)+\log_a(y)$)

\[\log_a(xy^{-1})=\log_a(x)+\log_a(y^{-1})\]

Next we can use the logarithmic power rule (Power Rule: $\log_a(x^r)\rightarrow r\log_a(x)$) to rewrite the following

\[\log_a(y^{-1})=(-1)\log_a(y)\]

Substituting this back in we have

\[\log_a(xy^{-1})=\log_a(x)+(-1)\log_a(y)\]

This gives us the logarithmic quotient rule

\[\log_a\left(\frac{x}{y}\right)=\log_a(xy^{-1}) =\log_a(x)-\log_a(y)\]

Watch this section: using known rules to get the quotient rule at 0:26.

Expanding Quotients ($\rightarrow$ direction)

Consider

\[\log_2\left(\frac{x+1}{z}\right)\]

We can rewrite this as a logarithm of the same base containing the numerator ($x+1$) subtracting the logarithm containing the denominator ($z$).

\[\log_2\left(\frac{x+1}{z}\right)=\log_2(x+1)-\log_2(z)\]

Rapidly applying it to another example

\[\log_5\left(\frac{10}{x-2}\right) \rightarrow \log_5(10)-\log_5(x-2)\]

Watch this section: expanding quotients at 1:54 and example expansion at 2:00.

Combining Differences ($\leftarrow$ direction)

Consider

\[\log_{\pi}(x)-\log_{\pi}(100)\]

We can rewrite the difference of two logarithms that have the same base by combining them using the quotient rule in the left direction.

\[\log_{\pi}(x)-\log_{\pi}(100)=\log_{\pi}\left(\frac{x}{100}\right)\]

Rapidly applying it to another example

\[\log_{11}(\pi)-\log_{11}(150+z)\rightarrow \log_{11}\left(\frac{\pi}{150+z}\right)\]

Watch this section: combining logarithms with the quotient rule at 2:08.

Timestamp Guide

Section	What is shown	Video
Quotient rule	Introduce $\log_a\left(\frac{x}{y}\right)=\log_a(x)-\log_a(y)$.	0:00
Rule derivation	Use negative exponents, the product rule, and the power rule.	0:26
Expanding and combining	Move between fractions inside logs and differences of logs.	1:54
Expansion examples	Expand $\log_2\left(\frac{x+1}{z}\right)$ and $\log_5\left(\frac{10}{x-2}\right)$.	2:00
Combining examples	Combine same-base log differences into one quotient.	2:08
Reciprocal rule	Show $\log_a\left(\frac{1}{x}\right)=-\log_a(x)$.	2:12
Closing	Comment, like, and subscribe to support the channel.	2:48

Use the logarithmic quotient rule topic page for more quotient rule notes. The logarithmic quotient rule example archive collects the timestamped examples from this video and related JoeCMath logarithm practice.

For nearby rules, review the logarithmic product rule companion, the logarithmic product rule archive, or the logarithmic power rule archive.

You can also watch the JoeCMath logarithms playlist: open the playlist on YouTube.