andydude Wrote:I'm going to re-write your equation in the form:
\( {}^{x}{a} = \lim_{n \rightarrow \infty} \log_a^{[n]}(\exp_b^{[n]}({}^{x+\mu_b(a)}{b})) \)
Using this form of your equation, as \( n \rightarrow \infty \) then \( \exp_b^{[n]}(\cdots) \) becomes \( {}^{\infty}b \) and \( \log_a^{[n]}(\cdots) \) becomes \( {}^{-\infty}a \). This then implies:
\( {}^{x}{a} = {}^{-\infty}a \)
which is strictly not true. If you want to make a change-of-base formula for tetration, at least make one that is consistent. This one is not. I've spent a great deal of time looking for a change-of-base formula, and I'm convinced that one does not exist. It could be that I forgot something about the limit process, and the simplifications above do not occur, we'll have to investigate in more detail.
Andrew Robbins
I think where your simplifications break down is that you're taking an infinite iterated logarithm of an infinite iterated exponential, but your not taking into account that your iterating each exactly the same number of times (edit: aside from the additive superlogarithmic constant).
I don't have time this morning to show why this matters, but the math I provided above should have been sufficient to point this out. For any finite n, the formula works, and it gets more and more and more accurate as n goes up, so why do you expect it to suddenly stop working when the limit goes to infinity?
