Prove that the function

let $x_{1}, x_{2} \in(0, \infty)$

We have, $x_{1}

$\Rightarrow \log _{e} x_{1}<\log _{e} x_{2}$

$\Rightarrow f\left(x_{1}\right)

So, $f(x)$ is increasing in $(0, \infty)$