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.
\({{\log }_{a}}b+{{\log }_{a}}c={{\log }_{a}}\left( b\times c \right)\) 
e.g. \(\ln 2+\ln5=\ln10\) , \({{\log }_{3}}2+{{\log }_{3}}20={{\log }_{3}}40\), 
\({{\log }_{2}}3+{{\log }_{2}}7+{{\log }_{2}}2={{\log }_{2}}42\)
Important notes:
The formula \({{\log }_{a}}(p+q)= \left( \log_ap \right)\left( \log_aq \right)\) is not an identity. 
The formula \({{\log }_{a}}(p+q)= \log_ap+\log_aq\) is not an identity.

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.

\({{\log }_{a}}b+{{\log }_{a}}c={{\log }_{a}}\left( b\times c \right)\) 









e.g. \(\ln 2+\ln5=\ln10\) , \({{\log }_{3}}2+{{\log }_{3}}20={{\log }_{3}}40\), 

\({{\log }_{2}}3+{{\log }_{2}}7+{{\log }_{2}}2={{\log }_{2}}42\)

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

Product Law

Product Law