Quotient Law and Its Proof

Quotient Law

\({{\log }_{a}}b-{{\log }_{a}}c={{\log }_{a}}\left( \frac{b}{c} \right)\)

e.g. \(\ln2 -\ln5 =\ln\frac{2}{5}\) , \(\log_32 - \log_320 =\log_3\frac{1}{10}\),
\(\log_216 - \log_22 =\log_2\frac{16\times2}{5}\)

Important note:

\(\log_a(p-q) \neq\frac{\log_ap}{\log_aq}\), \(\log_a(p+q) \neq \log_ap +\log_aq\)

Because logarithms are indices, the index laws translate into three log laws. They apply only when every logarithm has the same base and every argument is positive.

The quotient law applies to a quotient inside one logarithm: \(\log_a(M/N)=\log_a M-\log_a N\). It does not turn \(\log_a(M-N)\) into a quotient or a difference of logarithms.