We have

\(\displaystyle{\ln{:}}{\left({0},\infty\right)}\rightarrow\mathbb{R}\)

and \(\displaystyle{\ln{:}}{\left({1},\infty\right)}\rightarrow\mathbb{R}^{+}-{\left\lbrace{0}\right\rbrace}\)

\(\displaystyle{\ln{:}}{\left({0},\infty\right)}\rightarrow\mathbb{R}\)

and \(\displaystyle{\ln{:}}{\left({1},\infty\right)}\rightarrow\mathbb{R}^{+}-{\left\lbrace{0}\right\rbrace}\)