Change of base formula for Tetration

[sheldonison]

???

how does change of base formula for tetration and exp(z) -1 relate ???

i would recommend -- in case this is important and proved -- that more attention is given to it on the forum and/or FAQ

I agree that the base conversion problem is very interesting, and needs more attention.

Here's another way to write the change of base equation, converting from base a to base b.
\( \text{sexp}_b(x) =
\text{ } \lim_{n \to \infty}
\text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n))) \)

In this equation, \( \text{slog}_a(\text{sexp}_b(n))) \) converges to n plus the base conversion constant. This base conversion will have a small 1-cyclic periodic wobble, \( \theta(x) \), when compared to Dimitrii's solution.
\( \text{sexp}_b(x+\theta(x)) =
\text{ } \lim_{n \to \infty}
\text{log}_b^{\circ n}(\text{sexp}_a (x + \text{slog}_a(\text{sexp}_b(n))) \)

Convergence for real values of x is easy to show, and emperically the derivatives appear continuous, but behavior for complex values is a more difficult problem. I would also like to characterize the \( \theta(x) \) sinusoid, and find out whether or not it is c-oo, and whether the sexp_b(z) shows the singularities in the complex plane predicted by Dimitrii Kouznetsov. My thoughts started before I read Jay's post, but you can see them here, http://math.eretrandre.org/tetrationforu...hp?tid=236.