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 Change of base formula for Tetration jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/12/2007, 09:30 PM bo198214 Wrote:I believe that you have a lot of good ideas, but actually I can not quite follow your explanations. Most of your formulas comes without any justifications nor proofs. That your formula ${}^x a = \lim_{n \to \infty} \log_a^{\circ n}\left({}^{\left( n+x+\mu_b(a)\right)} b\right)$ yields a proper tetration the resulting ${}^xa$ must at least satisfy the identities $^{x+1}a=a^{{}^xa}$ and ${}^1 a=a$ if ${}^xb$ satisfies them. Can you verify this first?Sorry, I thought it was implicit in the formula, but I've been looking at this for several days, so perhaps I just took it for granted: $\begin{eqnarray} {}^x a & = & \lim_{n \to \infty} \log_a^{\circ n}\left({}^{\left( n+x+\mu_b(a)\right)} b\right) \\ \vspace{15} \\ {}^{x+1} a & = & \lim_{n \to \infty} \log_a^{\circ n}\left({}^{\left( n+(x+1)+\mu_b(a)\right)} b\right) \\ & = & \lim_{n \to \infty} \log_a^{\circ n}\left({}^{\left( (n+1)+x+\mu_b(a)\right)} b\right) \\ & = & \lim_{n \to \infty} {\Large a}^{\left[ \log_a^{\circ (n+1)}\left({}^{\left( (n+1)+x+\mu_b(a)\right)} b \right) \right]} \\ & = & \lim_{n \to \infty} {\Large a}^{\left[ \log_a^{\circ (n-1)}\left( \log_a \left({}^{\left( (n+x+\mu_b(a)\right)} b \right)+\epsilon_1 \right) \right]} \\ & \approx & \lim_{n \to \infty} {\Large a}^{\left[ \log_a^{\circ n}\left({}^{\left( (n+x+\mu_b(a)\right)} b \right) \right]} \\ & \approx & \lim_{n \to \infty} {\Large a}^{\left[{}^{x} a\right]} \\ \end{eqnarray}$ The base case ${}^1 a=a$ is guaranteed by the definition of $\mu_b(a)$. If you run the formula and you don't get ${}^1 a=a$, then your constant $\mu_b(a)$ needs to be adjusted. « 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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