Change of base formula for Tetration

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Change of base formula for Tetration

jaydfox $Offline$
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Posts: 440
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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.

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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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