What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$?
Is it $slog^{x}_{2}$?
What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$?
Is it $slog^{x}_{2}$?
The inverse function of ${^x{2}}$ is the base-2 super-log of x, which is what you wrote. The inverse function of ${^2{x}}$ is the 2nd super-root of x.
While the super-log seems to have a more commonly accepted notation (slog), there is no such notation for super-root (or, if you wish, dozens of notations). There are also some places (like Wikipedia's Tetration page) that call the 2nd super-root the "super-sqrt" function. But that's OK, because lots of things in mathematics don't have unique notations. Take $e$ for example. It is used for basis vectors, physical constants, dimensional units, other variables, and the "base of the natural logarithm".
So if you're looking for terminology, the inverse of ${^2{x}}$ is called "super-root", but if you're looking for notation, just use $f(x)$, along with the explanation: "where $f$ is the second super-root".