Question

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

Infinity

$2$

$\sqrt[4]{2}$

$\sqrt2$

None of these

Answer: D

Concepts:exponent recursion

Difficulty rating: 1830

Small Hint:

The exponent above the first $x$ is the entire infinite tower again

Big Hint:

Replace that repeated tower by its given value $2$

Solution:

Let the value of the infinite tower be $T.$ Removing its bottom $x$ leaves the same tower as the exponent, so $T=x^T.$ Since $T=2,$ we obtain $2=x^2.$ The positive base is therefore $x=\sqrt2,$ for which the tower is convergent.

Thus, the correct answer is D.

1950 AMC 12 Problem 42

Answer: D

Difficulty rating: 1830

Small Hint:

Big Hint:

Solution:

Problem 42 in Other Years