Infinite power towers

A few days ago I read an interesting problem on Hacker news:

Find $x$ such that the infinite power tower $x^{x^{.^{.^{.^x}}}}$ is equal to 2.

The solution relies just on simple algebra and the usage of logarithm rules. We start as follows:

$x^{x^{.^{.^{.^x}}}} = 2$

Proceed by taking the logarithm of both sides:

${x^{.^{.^{.^x}}}} \cdot \log{x} = \log{2}$

But ${x^{.^{.^{.^x}}}} = 2$ , so

$\\ 2 \cdot \log{x} = \log{2} \\ \log{x} = \log{2} \cdot \frac{1}{2} \\ \log{x} = \log{2^{\frac{1}{2}}} \\ \log{x} = \log{\sqrt{2}} \\ x = \sqrt{2}$

This also holds for the general case $x^{x^{.^{.^{.^x}}}} = k$ , where $x = \sqrt[\leftroot{-2}\uproot{2}k]{k}$ . You can try showing it yourself. 🙂

Published by Boro Sitnikovski