# LambertW function homework

1. Oct 24, 2014

### arpon

If, $f(x)=x^x$, then, f-1(x)=?

2. Oct 24, 2014

### guysensei1

I don't think there's a closed form expression for that.

3. Oct 24, 2014

### Staff: Mentor

4. Oct 27, 2014

### JJacquelin

5. Oct 27, 2014

### HallsofIvy

Staff Emeritus
Nicely done! I misread your answer at first and thought you had it wrong but saw that you are correct after working it out for myself. The Lambert W function is inverse to $f(x)= xe^x$ but taking the logarithm of both sides of $x^x= y$ gives $xln(x)= ln(y)$ not $xe^x= y$.

Instead, once you have $xln(x)= ln(y)$, let $u= ln(x)$. Of course, then, $x= e^{ln(x)}= e^u$ so the equation becomes
$ue^u= ln(y)$, $u= ln(x)= W(ln(y))$ so that, as you say, $x= exp(W(ln(y))$.