Finding the unknown variable What is the value of $x$ in $x^{x}=25$? How can this be solved in the easiest way of all? I just couldn't deduce any idea regarding where to start.
$$
\begin{align}
x^x&amp;=25\tag{1}\\
x\log(x)&amp;=\log(25)\tag{2}\\
\log(x)e^{\log(x)}&amp;=\log(25)\tag{3}\\
\log(x)&amp;=\mathrm{W}(\log(25))\tag{4}\\
x&amp;=e^{\mathrm{W}(\log(25))}\tag{5}\\
&amp;=\frac{\log(25)}{\mathrm{W}(\log(25))}\tag{6}
\end{align}
$$
Explanation:
$(2)$: take log of both sides
$(3)$: $x=e^{\log(x)}$
$(4)$: if $we^w=x$, then $w=\mathrm{W}(x)$, $\mathrm{W}$ is Lambert W
$(5)$: exponentiate $(4)$
$(6)$: divide $(2)$ by $(4)$
\begin{align}
x^x &amp;= 25 \\  
x^2 &amp;= 25
\end{align}
possibilities, $x=5$ or $x=-5$ because $5^2 = 25$ and $-5^2 =25$
description:
\begin{align}
-5\times-5  
&amp;=(-1)\times5\times(-1)\times5 \\
&amp;=(-1)\times(-1)\times5\times5 \\
&amp;=1\times5\times5 \\
&amp;=25
\end{align}