tetration base conversion, and sexp/slog limit equations

[sheldonison]

On the other hand Dmitrii's plot of the derivatives looks quite like yours:
[img
....
So can it just be a rounding error (Taylor series)?
And why not ask Dmitrii, to compute \( \text{slog}_{1.45}(x)-\text{slog}_{e}(x) \) for \( x \) to infinity, whether it wobbles or not

Dmitrii, can you do this?

No, its not a rounding error..... I'm not sure how the higher order derivatives will behave as the base approaches \( \eta \), but the lower derivatives should behave similar to Dimitrii's result.

The correct equation to have Dimitrii verify has 1-cyclic convergence as x increases is \( \text{slog}_{1.45}(\text{sexp}_{e}(x)) -x \). For integer values of x, I calculated a base conversion constant of 40.036534.

The wobble is not particular large, and has a peak to peak range of about 0.0008. Also, the wobble for base 1.45 is not all that different than the wobble Dimitrii would calculate for any other pair of bases, using his sexp/slog. Its just that a base like 1.45 has fewer degrees of freedom, so the \( \text{sexp}_{1.45} \) equations derived using my base conversion equation from Dimitrii's \( \text{sexp}_{e} \) equations already show the wobble in the first derivative of the \( \text{sexp}_{1.45} \) equation.

By the way Jay noticed the wobble in a previous post, which makes me wonder how the higher order derivatives will behave, when converting from a base approaching \( \eta \). So far, the 1st, 2nd and 3rd derivatives look well behaved, but after that, I'm outside of the error range of my calculations.

....
Which means that my change of base formula is on the right track, but for whatever reason it requires a cyclic shifting constant, a constant which "knows" the underlying exponent.

It'd be like saying that \( 4^x=2^{x\log_2(4,x)} \), where log_2(4,x) is no longer a constant, but a function of x that is cyclic though very nearly constant. Absurd!

But if I had to choose between this absurdity and denying the beauty of Andrew's solution, I'll take the absurdity. The superlogarithmic constant would appear to be a function of the underlying tetrational exponent, cyclic though very nearly constant....