08/15/2007, 11:41 PM

bo198214 Wrote:jaydfox Wrote:How do you figure that it only works for a<b? It works for any two bases greater than eta, regardless of relative size.Did you try it?

The sequence is decreasing for (you can even prove this, similar to my proof that it increases for ) and after some steps it becomes negative and after that the logarithm yields complex numbers. Its quite obvious also from the experimentation.

And I would expect real numbers from a change of base, as the tetration is real for real arguments.

You're not factoring in the superlogarithmic constant. You tetrate base b n+mu times, then take the logarithm base a only n times. If base b is greater than base a, then mu will be negative.

So, for example, if mu_b(a) is -1, and we use n=20, then you'll tetrate base b 19 times, then take 20 logarithms base a, and you'll be back at 1 (and some small epsilon value that is essentially nil).

Tetrate base b 20 times and take 20 logarithms base a, and you'll be back at a. Tetrate base b 21 times and take 20 logarithms base a, and you'll be at a^a. Tetrate base b 22 times, then 20 log_a's, and you'll be at a^^3. And so on. Going the other way, tetrate base b 18 times, then 20 log_a's, and you'll arrive at 0.

~ Jay Daniel Fox