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Superlogarithm is quite ambiguous term.

Superlogarithm may refer to solution $f$ of the transfer equation with transfer function logatirithm, id est


The reasonable solution of this equation can be easily expressed through the natural tetration tet,


Also, Superlogarithm may refer to the natural ArcTetration $\mathrm{ate}=\mathrm{tet}^{-1}$, reminding that it is inverse function of SuperExponential. This analogy had been used choosing names fsexp.cin and fslog.cin for the C++ numerical implementations of the natural tetration and the ArcTetration.

The ambiguity makes term Superlogarithm unwanted; in future, it is better to avoid it at all in any of its meanings. In particular, as soon as the more advanced algorithms for the tetration and the ArcTetration will be implemented, it is better to call them simply tet and ate, in the similar way, as the names of routines for elementary functions coincide with the names of these functions.