Logarithmic Quotient Rule
This page follows the JoeCMath video Logarithmic Quotient Rule in 3 Mins. The video shows why dividing inside one logarithm turns into subtraction, then uses the rule to expand and combine logarithms.
Quotient Rule For Logarithms
The video starts with the logarithmic quotient rule:
\[\log_a\left(\frac{x}{y}\right)=\log_a(x)-\log_a(y).\]The rule applies under the usual logarithm conditions shown in the video:
\[x>0,\qquad y>0,\qquad a>0,\qquad a\ne 1.\]The main idea is that a quotient inside one logarithm becomes a difference of two logarithms with the same base.
Watch this section: quotient rule for logarithms at 0:00.
Why The Rule Works
The video builds the quotient rule from rules already on the screen instead of presenting it as a disconnected formula.
Start with
\[\log_a\left(\frac{x}{y}\right).\]Rewrite the division using a reciprocal:
\[\frac{x}{y}=x\left(\frac{1}{y}\right).\]Then use a negative exponent:
\[\frac{1}{y}=y^{-1},\]so
\[\log_a\left(\frac{x}{y}\right)=\log_a(xy^{-1}).\]Now apply the logarithmic product rule:
\[\log_a(xy^{-1})=\log_a(x)+\log_a(y^{-1}).\]Finally, use the logarithmic power rule:
\[\log_a(y^{-1})=(-1)\log_a(y).\]That gives
\[\log_a\left(\frac{x}{y}\right)=\log_a(x)-\log_a(y).\]Watch this section: using known rules to get the quotient rule at 0:26.
Expanding Quotients
The video then uses the quotient rule to expand logarithms that already contain fractions.
For expansion, the numerator becomes the positive logarithm and the denominator becomes the logarithm being subtracted.
For example,
\[\log_2\left(\frac{x+1}{z}\right)=\log_2(x+1)-\log_2(z).\]The same pattern applies to
\[\log_5\left(\frac{10}{x-2}\right)=\log_5(10)-\log_5(x-2).\]Both examples use the same base before and after expansion. The quotient rule changes the quotient into subtraction; it does not change the logarithm base.
Watch this section: expanding quotients at 1:54 and example expansion at 2:00.
Combining Differences
The video also runs the rule in the other direction. If two logarithms have the same base and are being subtracted, they can be combined into one logarithm with a quotient.
For the first combining example,
\[\log_{\pi}(x)-\log_{\pi}(100)=\log_{\pi}\left(\frac{x}{100}\right).\]For the second example,
\[\log_{11}(\pi)-\log_{11}(150+z)=\log_{11}\left(\frac{\pi}{150+z}\right).\]The first argument becomes the numerator, and the second argument becomes the denominator.
Watch this section: combining logarithms with the quotient rule at 2:08.
Reciprocal Rule
Near the end, the video gives a quick overview of the logarithmic reciprocal rule:
\[\log_a\left(\frac{1}{x}\right)=-\log_a(x).\]The reciprocal rule is the quotient-rule idea compressed into the special case where the numerator is one.
The video also shows the negative-exponent reason:
\[\frac{1}{x}=x^{-1}.\]So
\[\log_a\left(\frac{1}{x}\right)=\log_a(x^{-1})=(-1)\log_a(x)=-\log_a(x).\]Watch this section: quick overview of the reciprocal rule at 2:12.
Timestamp Guide
Related Logarithm Work
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.