Crazy infinite exponents equation

Note that $\large \color{#D61F06} x^{x^{x^{x^{\cdot^{\cdot^\cdot}}}}} = 2$ . Therefore, $\large x^{\color{#D61F06}x^{x^{x^{\cdot^{\cdot^\cdot}}}}} = x^{\color{#D61F06}2} = 2$ , $\implies \large x = \boxed{\sqrt 2}$ .

Note that the solution is only valid if $e^{-e} \le x \le e^\frac 1e$ , which $\sqrt 2$ is. Refer to equation 13 of Power Tower .

Let $\Huge{ x }^{ { x }^{ { \dots }^{ x } } }=2=a$ By substituting a we get $\ {x}^a=2$ because a is 2 we get $\ {x}^2=2$ then $\ x= \sqrt{2}$