# What is the domain of arcsin(Ln(x))?

Let $y = \arcsin \left(\ln x\right)$ hence $\sin y = \ln x$
But the domain of $\ln x$ is $x > 0$
The domain of siny is $- 1 \le \sin y \le 1$ hence
$- 1 \le \ln x \le 1$ hence ${e}^{-} 1 \le x \le {e}^{1}$
Finally the domain of $y = \arcsin \left(\ln x\right)$ is ${D}_{y} = \left[\frac{1}{e} , e\right]$