Is there another representation for $x^x$

Question

I started wondering about this the other day. Since the following have their own alternate representations. $$\begin{align*} \displaystyle\large x+x=2x & \ \frac{x}{x}=1 & xx=x^2\end{align*}$$

Can $x^x$ be represented in some other way? Thanks.

Answer 1

Yes, it's called tetration. We can write $x^x$ as $^{2}x$.

There's actually a whole chain of these iterated operators, such as the (rather) famous Knuth up-arrow notation. The page I linked to has quite a few examples if you are interested.

Answer 2

$$e^{x\log(x)}$$

This is a nice way to represent it if you want to differentiate it, since you can then just apply the standard differentiation rules.

Something like $$x^{x^x}$$ will be represented as $$e^{e^{x\log(x)}\log(x)}$$