Super-logarithm

In mathematics, the super-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:

As the Abel function of exponential functions,
As the inverse function of tetration with respect to the height,
As a generalization of Robert Munafo's large number class system,

For positive integer values, the super-logarithm with base-e is equivalent to the number of times a logarithm must be iterated to get to 1 (the Iterated logarithm). However, this is not true for negative values and so cannot be considered a full definition. The precise definition of the super-logarithm depends on a precise definition of non-integer tetration (that is, ${^{y}x}$ for y not an integer). There is no clear consensus on the definition of non-integer tetration and so there is likewise no clear consensus on the super-logarithm for non-integer inputs.

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