The function $\exp:{\mathbb{R}}\rightarrow{\mathbb{R}}^{+}$ given by $y=e^{x}$ has an inverse, which we call the natural logarithm function, denoted by $\log_{e}$ or simply by $\log$.

Remarks (i) Napier used natural logarithms to help calculate products, since the logarithm converts products into sums.

(ii) On some calculators, $\ln$ is used for the logarithm function, to emphasize that natural logs to base $e$ are involved.

(iii) There are logs to other bases. If we take $y=10^{x}$, then the corresponding inverse function is $x=\log_{10}y$, where $\log_{10}$ is the logarithm function to base $10$. In the days before calculators, tables of logarithms to base 10 were widely used in calculations, but these tables are now almost obsolete. Logarithms to base 2 are sometimes used in computer science for binary calculations.