We are given a fascinating recursive expression:

\[ x^{x^{x^{\cdot^{\cdot^\cdot}}}} = 2 \]

This infinite power tower, also known as a tetration, asks:

What value of \(x\) causes this tower to converge exactly to 2?

Step 1: Understand the Definition

Let:

\[ y = x^{x^{x^{\cdot^{\cdot^\cdot}}}} \]

Given that \(y = 2\), and because the tower is infinitely recursive:

\[ y = x^y \]

Step 2: Solve the Equation

Substitute \(y = 2\) into \(y = x^y\):

\[ 2 = x^2 \]

Solving for \(x\):

\[ x = \sqrt{2} \]

Step 3: Validate Convergence

Infinite power towers don’t always converge. For convergence, it’s known that:

  • The tower \(x^{x^{x^{\cdots}}}\) converges if and only if \(0 < x \leq e^{1/e} \approx 1.444\ldots\)

Check if \(\sqrt{2} \approx 1.4142\) is within that range — it is.

Thus, the tower converges and the result is valid.

Final Answer

\(\boxed{x = \sqrt{2}}\)

Reference