Change of base formula for Tetration

[tommy1729]

???

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.

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 )

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

that isnt answered ...


furthermore i had the idea that

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

and

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

( derivative with respect to x )

and that this might require a different Coo slog but would imply a uniqueness condition ?

also the equation lim slog_a(oo) - slog_b(oo) = x

seems intresting.

regards

tommy1729