$x^{x^{x}}=2$ : How can I solve this exactly?

Asked Mar 28 '24 at 16:34

Active Mar 28 '24 at 16:34

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I solved the following equation today for fun and I have the exact real result:

$x^{x^{x+1}}=2 \Leftrightarrow x=\frac{W(ln(2))}{W(W(ln(2)))}=\exp ^{W(W(ln(2)))}$

Where $W$ is the Lambert W function.

But how can I now solve $x^{x^{x}}=2$ exactly? I know it's another similar question, but I just love exact results from such cool equations. Thanks for answers!

edited Mar 28 '24 at 16:34

asked Mar 28 '24 at 16:34

Mordor_07

See here. – Dietrich Burde Mar 28 '24 at 16:36
This question has an integral representation for the constant – Тyma Gaidash Mar 28 '24 at 16:48
@DietrichBurde I think so. It's just sad that you can't express it exactly in general, so as a result – Mordor_07 Mar 28 '24 at 16:50

$x^{x^{x}}=2$ : How can I solve this exactly?

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