Super-logarithm

In mathematics, the super-logarithm (or tetra-logarithm) is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions, roots and logarithms, tetration has two inverse functions, super-roots and super-logarithms. There are several ways of interpreting super-logarithms:

The precise definition of the super-logarithm depends on a precise definition of non-integral tetration (that is, ${\displaystyle {^{y}x}}$ for y not an integer). There is no clear consensus on the definition of non-integral tetration and so there is likewise no clear consensus on the super-logarithm for non-integer range.

The super-logarithm, written ${\displaystyle \,\mathrm {slog} _{b}(z)}$, is defined implicitly by

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