08/13/2007, 02:47 PM
I'm going to re-write your equation in the form:
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:
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
\( {}^{x}{a} = \lim_{n \rightarrow \infty} \log_a^{[n]}(\exp_b^{[n]}({}^{x+\mu_b(a)}{b})) \)
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
