Notation for the solution to $y^y=x$. Let $x$ be a positive real number.


*

*The solution to $y+y=x$ is written
$$y=x/2.$$

*The solution to $y\cdot y=x$ is written
$$y=x^{1/2}.$$

*Is there a notation for the solution to $y^y=x$?

 A: There's a well known function called the Lambert W function, defined to be the inverse of $xe^x$. If $y^y = x$, then
$$\ln(y)e^{\ln(y)} = y\ln(y)=\ln(x) \implies$$
$$\ln(y) = W(\ln(x)) \implies$$
$$y = e^{W(\ln(x))}$$
I don't know of any function simply defined to be the inverse of $x^x$, though, but problems like this can often be solved with the W function. 
Note: $xe^x$ and $x^x$ aren't injective on $(0,\infty)$, so you have to be careful about the possibilities of multiple solutions to these equations. 
A: Take logs:
$$y\ln y=\ln x=\ln y\cdot e^{\ln y}$$
The definition of the Lambert W function satisfies
$$W(\ln x)e^{W(\ln x)}=\ln x$$
Therefore
$$W(\ln x)=\ln y$$
$$y=e^{W(\ln x)}=\frac{\ln x}{W(\ln x)}$$
Any expression that relates an exponential of a variable with a linear function of the same variable is amenable to solution via Lambert W. Indeed, such expressions pop up in many physics equations, especially those relating to quantum mechanics.
A: As other's have stated, the $W$ function yields the correct answer to the question you asked.  A related sequence is this:
$$2+y=x\implies y=x-2$$
$$2\cdot y=x\implies y=x/2$$
$$2^y=2\uparrow y=x\implies y=\log_2(x)$$
$$\underbrace{2^{2^{2^{\cdot^{\cdot^\cdot}}}}}_{y}= 2\uparrow \uparrow y=x\implies y=\text{slog}_2(x)$$
where $\text{slog}$ is a superlog:
https://en.wikipedia.org/wiki/Super-logarithm
