Since $x^x$ grows very fast, its inverse should accordingly grow very slow, possibly slower than $\ln(\ln(x))$. I am troubled with finding such an inverse: I only get to the point: $\ln(x)x=\ln(y)$ after taking the logarithm of both sides.
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http://mathforum.org/library/drmath/view/70483.html – Cheerful Parsnip May 18 '15 at 15:32
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http://en.wikipedia.org/wiki/Lambert_W_function#Example_2 – Henry May 18 '15 at 15:34
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The function $x^x$ does not grow everywhere. But it does for $x\gt \frac{1}{e}$. Its logarithm grows only a little faster than $x$, so the inverse function, over the appropriate domain, grows defintely faster than $\ln(\ln x)$. – André Nicolas May 18 '15 at 15:51
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$$y=x^x\\ \ln y=x\ln x\\ \ln\ln y=\ln x+\ln\ln x=\ln x\left(1+\frac{\ln\ln x}{\ln x}\right)\\ \frac{\ln y}{\ln\ln y}=\frac{x}{1+(\ln\ln x/\ln x)} $$ The right-hand side is between $x$ and $x/2$, and the extra term in the denominator slowly tends to zero. So, for large $y$, the inverse function is near $\ln y/\ln\ln y$, and certainly between that value and $2\ln y/\ln\ln y$.
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