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 $f^{-1} (x) = e^{e^{e^x}}$