Change of base formula for Tetration

[sheldonison]

the problem is the wobble ...

its a bit of an illusionary use :

first you give a formula to compute sexp_b(x) bye using sexp_b(n)

then you correct sexp_b(x) to sexp_b(x + wobble(x))

which basicly just means ;
you got a formula for sexp_b(n) using sexp_b(n) ... ?!?

thats pretty lame selfreference ...

( godel escher and bach anyone ? :p )

Well, its not that bad, since n is an integer, sexp_b(n) is well defined. You can leave off the \( \theta(x) \) function, you just get a different solution, one that wobbles a little bit, easier to see in the higher derivatives. Also, in my original post, I was using a home base of \( \eta+\delta \) where \( \eta=e^{1/e} \), whose sexp solution I was able to derive, see http://math.eretrandre.org/tetrationforu...236&page=1.

furthermore , i asked how change of base formula for tetration and exp(z) - 1 relate ?

that isnt answered ...

Jay isn't around to answer. He discusses base change convergence, which I understand perfectly well. But I didn't understand the double logarithmic paragraph. Jay abandoned this approach to tetration, because it gives different results than Andrew Robbin's solution, (and Dimitrii Kouznetsov's solution) due to the wobble.

furthermore i had the idea that

slog_a(x) - slog_b(x) =/= 0 for a =/= b =/= x and a,b,x > e^e

For large enough values of x, slog_a(x) - slog_b(x) will converge to a specific value. That value will be the sexp base conversion constant plus the base conversion wobble term, \( \theta(\text{slog}_a(x)) \). Here are some examples of base conversion values I derived using sexp derived from base \( \eta^{+} \), which has a wobble when compared to Andy's solution or Dimitrii's solution.

\( \text{slog}_2(x) - \text{slog}_e(x) = 1.1282 \)
\( \text{slog}_3(x) - \text{slog}_e(x) = -0.1926 \)
\( \text{slog}_{10}(x) - \text{slog}_e(x) = -1.1364 \)