Solve the equation $x^{x^5} = 5$ over $\Bbb R.$ Find real numbers $x$ (if any) such that $x^{x^5} = 5.$ 

I have shown that $x \notin \Bbb Z.$ Does there exist any $x \in \Bbb R \setminus \Bbb Z$ which satisfies the above equation? Any help will be highly appreciated.
\begin{align} 
x^{x^5}&amp;=5
\tag{1}\label{1}
\end{align}

Equation \eqref{1} can be solved formally
with the help of the Lambert W function:

\begin{align} 
x^5\ln x&amp;=\ln 5
,\\
5x^5\ln x&amp;=5\ln 5
,\\
x^5\ln (x^5)&amp;=5\ln 5
,\\
\ln (x^5)\,\exp(\ln (x^5))&amp;=\ln 5\exp(\ln5)
.
\end{align}

At this point we can apply the Lambert W function:
\begin{align} 
\operatorname{W}(\ln (x^5)\,\exp(\ln (x^5)))
&amp;=
\operatorname{W}(\ln 5\exp(\ln5))
,
\end{align}
and since the argument 
$\ln 5\exp(\ln5)&gt;0$,
there is just one solution

\begin{align} 
\ln (x^5)&amp;=\ln5
,\\
x^5&amp;=5
,\\
x&amp;=5^{\tfrac15}\approx 1.379729661
.
\end{align}  



Edit

Or, less formally, without Lambert W function:

\begin{align} 
(x^{x^5})^5&amp;=5^5
,\\
x^{5x^5}&amp;=5^5
,\\
(x^5)^{(x^5)}&amp;=(5)^{(5)}
,\\
x^5&amp;=5
.
\end{align}
Here are graphs that show the solution, $x = \sqrt[5]{5}$: