Can you rearrange for $x$?

We have an equation $y=x^{x^x}$ . Can we rearrange for $x$ ?

So far I have managed to get it close to a form where the Lambert-W Function can be used.

$\frac{\log y}{x^{x-1}}\exp\left({\displaystyle\frac{\log y}{x^{x-1}}}\right)=x\log y$

However there is a pesky $x$ on the RHS which won't go away. I am wondering if the people of Brilliant.org have any idea on this?

If any body is curious, $x^x=y$ can be rearranged as $x=\frac{\log y}{W(\log y)}$ .

#Algebra #Rearrangement #ExponentialFunction #LambertW

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Can you rearrange for \(x\)?

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