College Algebra, Chapter 5, 5.2, Section 5.2, Problem 80

Supposed that $f(x) = \ln (\ln (\ln x))$. (a) Find the domain of $f$. (b) Find the inverse function of $f$.

a.) To determine the domain of $f$, we want $\ln (\ln x) > 0$


$\begin{equation} \begin{aligned} \ln (\ln x) >& 0 \\ \\ e^{\ln (\ln x)} >& e^0 \\ \\ \ln x >& 1 \\ \\ e^{\ln x} >& e^1 \\ \\ x >& e^1 \end{aligned} \end{equation}$


Thus, the domain of $f$ is $(e, \infty)$

b.) To find the inverse of $f$, we set $y = f(x)$


$\begin{equation} \begin{aligned} y =& \ln (\ln ( \ln x)) && \text{Solve for } x \\ \\ e^y =& e^{\ln (\ln ( \ln x))} && \\ \\ e^y =& \ln (\ln x) && \\ \\ e^{e^y} =& e^{\ln ( \ln x)} && \\ \\ e^{e^y} =& \ln x && \\ \\ e^{e^{e^y}} =& e^{\ln x} && \\ \\ e^{e^{e^y}} =& x && \text{Interchanging $x$ and $y$} \\ \\ y =& e^{e^{e^x}} && \end{aligned} \end{equation}$


Thus, the inverse of $f$ is $f^{-1} (x) = e^{e^{e^x}}$