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.
\[{{\log }_{a}}{{b}^{r}}=r{{\log }_{a}}b\]
e.g. \({{\log }_{2}}{{5}^{x}}=x{{\log }_{2}}5\) , \({{\log }_{3}}\sqrt{2}=\frac{1}{2}{{\log }_{3}}2\)
Important note: The formula \({{\left( {{\log }_{a}}x \right)}^{3}}= 3{{\log }_{a}}x\) is not an identity. But \({{\left( {{\log }_{a}}x \right)}^{3}}=\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)\)

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.

\[{{\log }_{a}}{{b}^{r}}=r{{\log }_{a}}b\]

e.g. \({{\log }_{2}}{{5}^{x}}=x{{\log }_{2}}5\) , \({{\log }_{3}}\sqrt{2}=\frac{1}{2}{{\log }_{3}}2\)

Important note:
The formula \({{\left( {{\log }_{a}}x \right)}^{3}}= 3{{\log }_{a}}x\) is not an identity. 
But \({{\left( {{\log }_{a}}x \right)}^{3}}=\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)\)

Power Law

Power Law