Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Possible Duplicate:
$x^x=y$. 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 $$x^{f(x)}=k$$

share|cite|improve this question

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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.

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.