1. ## functions(2)

given the functions $g(x)=e^x-2, x=(a,infinity)$
$
h(x)=ln(lnx), x>1
$

Find the smallest value of $a$ in exact form such that the composite function $hg$ exists.

2. Assuming that you are trying to find $\displaystyle h\left(g(x)\right)$...

$\displaystyle h\left(g(x)\right) = \ln{\left[\ln{\left(e^x - 2\right)}\right]}$.

A logarithm is only defined for positive values of the independent variable, so the innermost logarithm can only be evaluated where $\displaystyle e^x - 2 > 0$.

But since another logarithm will be taken, you can only accept those values for which the innermost logarithm is positive. So where $\displaystyle e^x - 2 > 1$.

Solve for $\displaystyle x$.