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

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Jameson

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Jan 26, 2012
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MarkFL

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Feb 24, 2012
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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:

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

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

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

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:

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