Question 115733

All real logarithms below have positive arguments and a positive base different from 1. In the power law, the argument before raising it to a power is positive. Statements labelled as common mistakes describe rules that are not identities; equality can still occur for particular allowed values.

Quotient Law:

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

e.g. \(\ln 2-\ln5=\ln\frac{2}{5}\) , \({{\log }_{3}}2-{{\log }_{3}}20={{\log }_{3}}\frac{1}{10}\), \({{\log }_{2}}16-{{\log }_{2}}5+{{\log }_{2}}2={{\log }_{2}}\frac{16\times 2}{5}\)

Important note:

The formula \({{\log }_{a}}(p-q)= \frac{\log_ap}{\log_aq}\) is not an identity. , The formula \({{\log }_{a}}(p+q)= \log_ap+\log_aq\) is not an identity.