# 1950 AHSME Problems/Problem 42

## Problem

The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to: $\textbf{(A)}\ \infty \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{2} \qquad \textbf{(D)}\ \sqrt{2} \qquad \textbf{(E)}\ \text{None of these}$

## Solution

Solution 1:

Taking the log, we get $\log_x 2 = x^{x^{x^{.^{.^.}}}}=2$, and $\log_x 2 = 2$. Solving for x, we get $2=x^2$, and $\sqrt{2}=x \Rightarrow \mathrm{(D)}$

Solution 2: $x^{x^{x^{.^{.^.}}}}=2$ is the original equation. If we let $y=x^{x^{x^{.^{.^.}}}}$, then the equation can be written as $y=2$. This also means that $x^y=2$, considering that adding one $x$ to the start and then taking that $x$ to the power of $y$ does not have an effect on the equation, since $y$ is infinitely long in terms of $x$ raised to itself forever. It is already known that $y=2$ from what we first started with, so this shows that $x^y=x^2=2$. If $x^2=2$, then that means that $\sqrt{2}=x \Rightarrow \mathrm{(D)}$.

This is just a faster way once you get used to it, instead of taking a log of the function.

~mathmagical

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