From AMS Glossary
The logarithm of any positive number n to the base b is the power l to which that base must be raised in order to satisfy the identity n = bl: l = logbn.
Logarithms to the base 10 are called common logarithms and written log or log10. Logarithms to the base e = 2.7182818284 . . . are called natural (Napierian, hyperbolic) logarithms, and are often written loge or ln. The natural logarithms are the more convenient in any computations involving differentiation