How to solve for $x$ in the function $y=x\ln(x)$?

Question

I have $y=x\ln(x)$ and I need to solve for x. How am I supposed to do that?

Because I get $y=x\ln(x)$

$y=\ln(x^x)$

$e^y=x^x$

and I am stack here...

Can somebody help me?

EDIT

In my case the result of y is always a real positive number (y>=1). Therefore, I would like to avoid any solutions that contain any imaginary part.

Answer 1

Your equation $z=x^x$ is solved here in terms of Lambert's W function. All you have to do is write x as $e^{\ln x}$, and then you have $\ln x=W\big(y\big)$, from which $x=e^{\ln x}=e^{W(y)}$ immediately follows.

Answer 2

The solution can be obtained as follows:

$$y=x\ln x=e^{\ln x}\ln x $$

This is the defining equation for the Lambert W function. Thus,...

$$\ln x=W(y)$$

$$x=e^{W(y)}$$

If $x=1$ then, trivially, $y=0$.

A pretty good explanation of this function can be found on wiki: http://en.m.wikipedia.org/wiki/Lambert_W_function