Answer
$f^{-1}(x)=\frac{log(log(x))-log(log2)}{log3}$
domain{$x|x\gt1$}
range {$y|y\in R$}
Work Step by Step
1. Given $y=f(x)=2^{3^x}$, take logarithm on both sides, we have
$log(y)=3^xlog2$, take logarithm again, we obtain:
$log(log(y))=log(3^xlog2)=xlog3+log(log2)$
Thus, $x=\frac{log(log(y)-log(log2)}{log3}$, switch $x,y$, we get the inverse function as
$f^{-1}(x)=\frac{log(log(x))-log(log2)}{log3}$
2. To determine the domain, let $log(x)\gt0$, we get $x\gt1$ which is {$x|x\gt1$}
The range would be {$y|y\in R$}
