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[jaydfox]

Those notations are useful for bases above eta. However, for bases between 1 and eta, you cannot recover the full exp function, unless you extend the definition of slog to include the 2 additional domains (only 1 additional for eta). While I'm in favor of such an extension, I'm still working on defining the correct z value to start from, just as 1 is the "correct" value to start from for the principal domain.

Until we define good starting points, a full analysis of fractional iterates will require the more general "T" notation, where we can start from a location other than 1 and leave it at that. For example, for base sqrt(2), we could start at sqrt( or e, etc., to solve for the corridor between the asymptotes at 2 and 4, and we could start at 4.1 or 5 or 16, etc. to solve above the asymptote at 4. (BTW, 16 seems to me the best place. Essentially, exponentiate the upper and lower asymptotes. The order doesn't matter: 2^4=4^2, and this is true for any base between 1 and eta, so it seems pretty "unique" to me.)

Ach, what am I saying? Even if we extended slog to include all 2 or three domains, the sexp function would cease to be a function if we tried to take the inverse of the slog. In some ways, this would indicate that the slog function may be more fundamental than the sexp function. Given that Andrew's solution is based on solving the slog, this is interesting...