1950 AHSME Problems/Problem 42

Problem

The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to:

$\textbf{(A)}\ \infty \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt[4]{2} \qquad \textbf{(D)}\ \sqrt{2} \qquad \textbf{(E)}\ \text{None of these}$

Solution

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 $\sqrt{2}=x \Rightarrow \mathrm{(D)}$

See Also

1950 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 41
Followed by
Problem 43
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