The Natural Logarithm

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $\exp:\mathbb{R}\to\mathbb{R}$ be the exponential function, and let $(0,\infty)$ be the ray of positive reals. By Basic Properties of the Exponential Function (xi), the map $E:\mathbb{R}\to(0,\infty)$, $u\mapsto\exp(u)$, is bijective.

(i) The natural logarithm is the map $\log=E^{-1}:(0,\infty)\to\mathbb{R}$, the inverse of $E$, which is a bijective map $(0,\infty)\to\mathbb{R}$ by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (vi).

(ii) $\exp(\log t)=t$ for every $t\in(0,\infty)$, and $\log(\exp u)=u$ for every $u\in\mathbb{R}$: by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (vi), $E\circ\log=\mathrm{id}_{(0,\infty)}$ and $\log\circ E=\mathrm{id}_{\mathbb{R}}$, and the values of these maps are given by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (ii) and Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (iii).