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I'm a little concerned that the article lacks a definition -- just the promise that it "presumably" can be continued. The function notation makes it look like there is a standard definition but there isn't. Worse the iterated logarithm is defined in terms of this (very) partial function, making it ill-defined. (What is ln*(400)? You need to find slog_e 400, but the article doesn't give you the means to calculate it.)

I think it would be better to be open about the state of development. There are lots of candidates for extending tetration to the real or complex numbers, and slog stands for the inverse of one such extension which must be made clear in context. (We don't have a Bohr-Mollerup theorem for tetration that gives us an obvious choice.) Then list properties that such a function is expected to have, and explain how log* is just the floor of any mototone function satisfying these conditions.

Charles R Greathouse IV 18:55, 13 May 2013 (UTC)

Over the next few days, I will work on the tetration functions tetra-powers and tetra-exponentials (and their inverses tetra-roots and tetra-logarithms) to give well-defined definitions and algorithms for the calculation over the real numbers (I don't know yet how to extend over the complex numbers...) I will add external links to support the articles. — Daniel Forgues 23:50, 13 May 2013 (UTC)
In that case be sure to mention that yours is not the only definition, and that others exist. I have a binder full of papers extending the tetration past the integers, and I don't think any two of the methods are alike... Charles R Greathouse IV 19:15, 14 May 2013 (UTC)
For now, it seems safe to consider tetration with nonnegative real numbers as base (otherwise complex numbers are involved) and nonnegative integers as height (the non-commutativity and non-associativity of exponentiation getting in the way of a naive generalization to positive rational numbers as height ). (The negative integers as height having a degenerate interpretation, that precludes all negative numbers and thus probably the complex numbers too...)
For the tetra-logarithm, since we need a domain were the tetration is strictly monotonic in the height , we will need as the base. — Daniel Forgues 04:21, 18 May 2013 (UTC)
Well, you actually need to take general tetra-logs. But this isn't the real problem. The real issue is that until you have a definition for tetration to non-integer heights you can't take tetra-logs except for a small fraction of numbers. Charles R Greathouse IV 04:34, 18 May 2013 (UTC)