Inverse function of $x^x$

Question

How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried using logs, differentiating, etc, etc, but to no avail.

Answer 1

You have to use the Lambert W function, the inverse of $x e^x$. Using this function, one can find that the inverse of $y=x^x$ is $x=e^{W(\ln(y))}$

Answer 2

As the other user mentioned, it is basically the application of Lambert W Function.

Say, $x^x = z$ which implies, $x \ln x = \ln z$.

Now, I can write: $x = e^{\ln x} $ using the properties of logarithms and exponential functions.

Therefore, $$\ln x = W \ln z \\ x = e^{W \ln z} $$

which is indeed the inverse of $x^x$ .

I suggest you to go through the Wikipedia's page for Applications of Lambert W Function.

Hope it helps!

Answer 3

To find the inverse of the function $ y=x^x$ (where it is well defined) you first have to switch $y$ with $x$ and viceversa and now you proceed this way:

$x=y^y$

$\ln x=y\ln y$

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

Now we apply Lambert's W function defined as:

$$W(z)e^{W(z)}=z$$

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

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