Quotient Law

For \(a>0,\ a\ne1\) and \(b,c>0\),

\(\log_a b-\log_a c=\log_a\left(\dfrac{b}{c}\right)\)

Example

\(\ln2-\ln5=\ln\dfrac25\)

\(\log_3 2-\log_3 20=\log_3\dfrac1{10}\)

\(\log_2 16-\log_2 5+\log_2 2=\log_2\left(\dfrac{16\times2}{5}\right)\)

Take note

The logarithms being combined must have the same base. The following common mistakes are not identities, even though equality may hold for particular allowed values.

The claim \(\log_a(p-q)=\dfrac{\log_a p}{\log_a q}\) is not an identity.

The claim \(\log_a(p+q)=\log_a p+\log_a q\) is not an identity.

Check that every logarithm argument is positive and every denominator is nonzero.