Is there a solution to the equation x^x^x^x^x^x^... = 2?

Question

I have been asked the following brainteaser, is there a solution to the equation:

$$ x^{x^{x^{...}}} = 2$$

(x to the power of itself an infinite number of times)

I am not sure about how to approach this one.

Answer 1

Let $x_1 = x$ and define $x_{n+1} = x^{x_n}$. If the limit exists, we then have $y=x^y$. Hence, $$2=x^2 \implies x = \sqrt2$$