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.

Suppose there is one, and find out what it must be. Then check whether it really is a solution. You probably interpret the power-tower as $x_1 = x,, x_{n+1} = x^{x_n}$ and then take $\lim_{n\to\infty} x_n$ as the value of $x^{x^{x^{\dotsc}}}$. — Daniel Fischer, Nov 11 '14 at 19:31

score 1 · Accepted Answer · answered Nov 11 '14 at 19:37

1

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

answered Nov 11 '14 at 19:37

Adhvaitha

1 Answers1