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 Change of base formula for Tetration jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/15/2007, 11:29 PM I've been reviewing my superlogarithmic constants, and realized I forgot a factor of log_b(a) when I was calculating, so, for example, $\mu_2^{\small -1}(1)$ is closer to 2.574062876128519046336549796971138669. What this means is that if you tetrate base 2 n times, and tetrate base 2.574062876128519046336549796971138669... only {n-1} times, then as n goes to infinity, these answers will converge. Here are the first few terms of each, using a=2, b=2.574062876128519046336549796971138669..., with $\mu_b(a)=-1$: $ \begin{array}{c|ccccc} n & {}^{(n-1)} b && {}^{n} a && \frac{{}^{(n-1)} b}{{}^{n} a} \\ \hline 0 & 0.00000000000000000000 && 1.00000000000000000000 && --- \\ 1 & 1.00000000000000000000 && 2.00000000000000000000 && 2.00000000000000000000 \\ 2 & 2.57406287612851904634 && 4.00000000000000000000 && 1.55396359471068572326 \\ 3 & 11.4014349335617688931 && 16.0000000000000000000 && 1.40333213259865135794 \\ 4 & 48044.9265758905624669 && 65536.0000000000000000 && 1.36405661681012200348 \\ 5 & 1.46881265672123\times 10^{19728} && 2.0035299304068\times 10^{19728} && 1.36404729441755212824 \\ 6 & 10^{10^{6.03122606263029537156455}} && 10^{10^{6.03122606263029537156455}} && 1.36404729441755212824 \\ \end{array}$ As you can see, the ratio goes to log_a(b) with increasing n. As it is, for n=6, the best I could do was to compare their logarithms. For row n=7, the discrepency in the logarithms would require about 19750 digits of accuracy, so for all practical intents and purposes, you would work with something in the neighborhood of n=6 when doing a base transformation between a and b (assuming you had an exact solution for either a or b). ~ Jay Daniel Fox « Next Oldest | Next Newest »

 Messages In This Thread Change of base formula for Tetration - by jaydfox - 08/12/2007, 06:39 AM RE: Change of base formula for Tetration - by jaydfox - 08/12/2007, 06:41 AM RE: Change of base formula for Tetration - by jaydfox - 08/12/2007, 08:38 AM RE: Change of base formula for Tetration - by jaydfox - 08/12/2007, 08:53 AM RE: Change of base formula for Tetration - by jaydfox - 08/12/2007, 04:34 PM RE: Change of base formula for Tetration - by bo198214 - 08/12/2007, 06:39 PM RE: Change of base formula for Tetration - by jaydfox - 08/12/2007, 09:00 PM RE: Change of base formula for Tetration - by jaydfox - 08/12/2007, 09:09 PM RE: Change of base formula for Tetration - by jaydfox - 08/12/2007, 09:30 PM RE: Change of base formula for Tetration - by andydude - 08/13/2007, 02:47 PM RE: Change of base formula for Tetration - by andydude - 08/13/2007, 02:58 PM RE: Change of base formula for Tetration - by jaydfox - 08/13/2007, 06:34 PM RE: Change of base formula for Tetration - by bo198214 - 08/13/2007, 07:31 PM RE: Change of base formula for Tetration - by bo198214 - 08/13/2007, 07:37 PM RE: Change of base formula for Tetration - by jaydfox - 08/13/2007, 08:03 PM RE: Change of base formula for Tetration - by bo198214 - 08/13/2007, 08:23 PM RE: Change of base formula for Tetration - by jaydfox - 08/13/2007, 08:32 PM RE: Change of base formula for Tetration - by bo198214 - 08/15/2007, 09:36 AM RE: Change of base formula for Tetration - by jaydfox - 08/15/2007, 10:15 AM RE: Change of base formula for Tetration - by jaydfox - 08/13/2007, 03:42 PM RE: Change of base formula for Tetration - by jaydfox - 08/15/2007, 11:29 PM RE: Change of base formula for Tetration - by jaydfox - 08/16/2007, 09:21 PM RE: Change of base formula for Tetration - by bo198214 - 08/16/2007, 09:32 PM RE: Change of base formula for Tetration - by jaydfox - 08/16/2007, 09:49 PM RE: Change of base formula for Tetration - by bo198214 - 08/16/2007, 09:53 PM RE: Change of base formula for Tetration - by jaydfox - 08/31/2007, 03:51 AM RE: Change of base formula for Tetration - by jaydfox - 08/31/2007, 06:24 AM RE: Change of base formula for Tetration - by bo198214 - 08/31/2007, 09:03 AM RE: Change of base formula for Tetration - by tommy1729 - 05/03/2009, 10:20 PM RE: Change of base formula for Tetration - by sheldonison - 05/05/2009, 11:21 AM RE: Change of base formula for Tetration - by tommy1729 - 05/05/2009, 12:19 PM RE: Change of base formula for Tetration - by sheldonison - 05/05/2009, 01:29 PM RE: Change of base formula for Tetration - by bo198214 - 05/05/2009, 01:37 PM RE: Change of base formula for Tetration - by jaydfox - 08/31/2007, 04:11 AM Convergence of Jay's change of base formula - by bo198214 - 08/15/2007, 09:00 PM RE: Parabolic Iteration - by jaydfox - 08/15/2007, 09:19 PM RE: Parabolic Iteration - by bo198214 - 08/15/2007, 09:30 PM RE: Parabolic Iteration - by jaydfox - 08/15/2007, 11:41 PM RE: Parabolic Iteration - by bo198214 - 08/16/2007, 08:17 AM RE: Parabolic Iteration - by jaydfox - 08/16/2007, 05:51 PM RE: Parabolic Iteration - by bo198214 - 08/16/2007, 06:40 PM RE: Parabolic Iteration - by jaydfox - 08/16/2007, 09:47 PM RE: Parabolic Iteration - by bo198214 - 08/16/2007, 10:07 PM

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