Relations and functions

Question

$$ g(x) = e^x-1, x\in\mathbb R\\ h(x) = \ln(\ln x),x>1\\ $$ By restricting the domain of $g$ to $(\alpha, \infty)$, where $\alpha \in \mathbb R$, find the smallest value of $\alpha$ in exact form such that the composite function $h\circ g$ exists. Define $h \circ g$.

My question: I don't understand why the restricted range of the g(x) is (1,∞)

Answer 1

$h\circ g(x)=h(g(x))=\ln(\ln(e^x-1))$ will be well defined if $\ln(e^x-1)$ is well defined and positive.

For this we need $e^x-1>1$ or equivalently $x>\ln2$.