tetration base conversion, and sexp/slog limit equations

[sheldonison]

Here I see some difficulties to put that mathematically. You want to define tetration only for one base, but "approaching" means that you have to define tetration at least for a sequence of bases.
But perhaps this is just a question of exactness. The more exact the tetration for arbitrary bases should be the closer to \( \eta \) one have to choose the base of the initial tetration, something in that direction.

OK, here's my proposal for a rigorous mathematical approach -- going all the way back to my original post:
> \( \text{slog}_2(x) - \text{slog}_e(x) = 1.1282 \)

This is equivalent to analyzing the following equation as b approaches \( \eta^+ \), for increasing values of n. Subtracting the two terms slog terms from each other cancels out the fact that slog(x) increases as the base approaches \( \eta^+ \). There are other ways to handle this, but this is a concise way to handle it in a limit equation. In these equation, slog\( _b \) refers to the slog with a linear approximation of the critical section.

\( \lim_{b \to \eta^+}\text{ } \lim_{n \to \infty} (\text{slog}_b(\text{sexp}_2(n)) - \text{slog}_b(\text{sexp}_e(n))) \)

First off, we can show that as n increases, the series converges for any individual value of b, (convergence as n increases is an easier problem discussed in the base conversion post by Jay; values of n>6 ought to give more or less unlimited accuracy for bases greater than 2).

The harder part is to show that the limit converges as b approaches \( \eta^+ \). If it converges, an extension of the sexp/slog function to real numbers can be defined. As an example of how this equation could define the sexp/slog function extension to real numbers, consider the following equation, where x is a real number. If the limit above converges, then the limit below should converge to x.

\( \lim_{b \to \eta^+}\text{ } \lim_{n \to \infty} (\text{slog}_b(\text{sexp}_e(n+x)) - \text{slog}_b(\text{sexp}_e(n)))=x \)

With a little additional algebra, this allows defining an sexp/slog extension to real numbers, for base e, by iterating the ln function "n" times. In practice, for base e, using n=5 will give approximately a million digits of precision for positive values of x. This also works for any other arbitrary base.

\( \text{sexp}_e(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty}
\text{ln(ln(ln(}\cdots
\text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n)))))) \)