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$x^x=y$. How to solve for $x$?

If we have $x^x=4$ it's easily solved by substituting $x$ with $2$. But for general equation like $x^x = k$, how we can find the solution?

Moreover, what if we have equation $$x^{a+bx}=k$$ or $$x^{f(x)}=k$$

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marked as duplicate by J. M., Gerry Myerson, Henning Makholm, Asaf Karagila, t.b. Nov 5 '11 at 22:17

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1 Answer 1

Solving $x^x=k$ requires the services of the Lambert function, $W(x)$. Recall that the Lambert function is the inverse of the function $xe^x$.

From this consideration, here's how to use the Lambert function:

$$\begin{align*}e^{x\ln\,x}&=k\\x\ln\,x&=\ln\,k\\e^{\ln\,x} \ln\,x&=\ln\,k\\\ln\,x&=W(\ln\,k)\\x&=e^{W(\ln\,k)}\end{align*}$$

and presto!

Your other equations are a bit more complicated; so far as I can tell they do not look like they have clean closed forms.

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