$X^X$ Algebra Problem

The solution can be given exactly only in terms of the Lambert W function, also called the “product logarithm”.

$\color{#3D99F6}{\displaystyle x = e^{W(2 (\log 2 + \log 5 ))}}$

In numerical form, $\color{#CEBB00}{\boxed{x \approx 3.597285023540417505497652251782286069136}}$

The W-function is defined so that $\color{#3D99F6}{W(ze^z) = z}$ on its primary branch. It can also be expressed using the Taylor series:

$\color{#D61F06}{\displaystyle W_0( x ) = \sum_{n=1}^{\infty} \frac{(-n)^{n-1}}{n!}x^n}$

but this converges only for $\color{#D61F06}{x < 1/e}$ , so the W function is the analytic continuation of this series. (That is, you can’t get a numerical value for the solution above using this series.)

There is also a family of complex solutions to the equation you give.