## Natural logarithm of real numbers

### Function

 definiendum $\mathrm{ln}:\mathbb R_+^*\to \mathbb R$ postulate $\mathrm{ln}=\mathrm{exp}^{-1}$

$\int_1^y \frac {1 } {x} {\mathrm d}x = \ln(y)$

$\int_0^{y} \frac {1 } {1+x } {\mathrm d}x = \ln(1+y)$

The function $x\mapsto\frac{x}{x-1}\log(x)$ is one without bad behaviours (singularities) on $[0,\infty)$.