Prove that the logarithmic function is strictly increasing on $(0,\infty )$

We have, $f(x) = \log x$ …(i)
(Note that, $\log x$ is defined only for $x > 0$)

Domain of $f{\rm{ }}\left( x \right)$ is $(0,\;\infty )$
Now, $f'(x) = \cfrac{1}{x} > 0$ for all $x \in (0,\;\infty )$

$\Rightarrow f'(x) > 0$ for all $x \in (0,\;\infty )$
$\therefore f$ is strictly increasing on $(0,\;\infty )$