What is Logarithm?

The rules follow from the rules of powers. $\log(ab) = \log a + \log b$. $\log\dfrac{a}{b} = \log a - \log b$. $\log a^n = n\log a$. The logarithm of $1$ is $0$ for every base. Base $10$ gives the common logarithm, used for pH, and base $e$ gives the natural logarithm, $\ln$. Logarithms turn multiplication into addition and make huge ranges of numbers manageable, as in the decibel and Richter scales.