Problem of the week #95 - January 20th, 2014

[Jameson]

Solve for x.

x^(x^(x^(x^(x^(x^(x^(x^(x…)))…) = 2
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Remember to read the POTW submission guidelines to find out how to submit your answers!

 

[MarkFL]

I am standing in for Jameson this week. He has been working on various coding projects around the clock and so I am giving him a well-deserved break from his POTW duties.

Congratulations to the following members for their correct solutions:

1) MarkFL
2) magneto
3) anemone
4) topsquark
5) mente oscura
6) Pranav
7) eddybob123
8) jacobi

Honorable mention goes to springfan25 for having only made a minor but critical error in the last step of the logarithmic method.

There were basically two methods used by those who submitted solutions. One was to use logarithms, as illustrated by topsquark:

Spoiler

\(\displaystyle x^{x^{x^x...}} = 2\)

\(\displaystyle \ln \left ( x^{x^{x^x...}} \right ) = \ln(2)\)

\(\displaystyle x^{x^{x^x...}} \cdot \ln(x) = \ln(2)\)

From the original problem statement \(\displaystyle x^{x^{x^x...}} = 2\) so
\(\displaystyle 2 \ln(x) = \ln(2)\)

etc., so \(\displaystyle x = \sqrt{2}\).

-Dan

The other was to use a substitution, as illustrated by magneto:

Spoiler

Let $p := x^{x^{x^{x^{\cdots}}}}$. We can rewrite the equation as $x^p = x^2 = 2$. Therefore, $x = \sqrt{2}$.