jaydfox Wrote:First, we need the constant of base conversion. It's essenstially a form of superlogarithmic constant. Think of it as the equivalent of the constant \( log_b(a) \) used for converting \( a^z = b^{z \times log_b(a)} \), assuming a and b are positive real numbers. We can find \( log_b(a) \) by taking a and b to very high integer powers:I should have trusted my first instinct. I called it a "superlogarithmic constant". As it turns out:
\( log_b(a)\ =\ \lim_{n \to \infty} \left(\frac{n}{k}\right),\ a^k \le b^n \le a^{k+1} \)
By analogy, for tetration, we're going to tetrate them each a large number of times. However, as you will see, tetration to integer powers won't work, not if we want to find the superlogarithmic constant. If the superlogarithmic constant isn't an integer, you can only approximate without an exact solution for one of the bases. In other words, in almost all cases, we must have an exact solution for fractional iteration for at least one of the bases. That doesn't mean the constant doesn't exist, only that we can't uniquely determine its value without an exact solution for some base.
\( \begin{array}{|ccccc|}& & & & \\
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\hspace{10} & {\Large ^{\normalsize x} a} & = & {\Large \lim_{n \to \infty} log_a^{\circ n}\left({}^{\left(\normalsize n+x+\mu_b(a)\right)} b\right)} & \hspace{10} \\
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\end{array} \)
In a twist of irony, the logarithmic constant for exponentiation (hyper-3) is multiplicative (hyper-2), but the superlogarithmic constant for tetration (hyper-4) is additive (hyper-1). And I say it's a "superlogarithmic" constant, but it should not be confused with \( slog_b(a) \). I think they're related, but I haven't pinned down the nature of the relationship yet. This will require more study.
\( \mu_e(b)\ =\ \lim_{z \to \infty} slog_e(z)-slog_b(z) \)
And there you have it. We've got an exact formula for base conversion of tetration, and an exact formula for finding the superlogarithmic constant. But these two facts together are only sufficient to solve for integer tetration and integer superlogarithms (i.e., where the slog_b(x) = n, n an integer).
We need only 1 exact solution to fill in all the gaps. But the solution must be unique. If we find "a" solution that is not "the" solution, then we get the wrong solution for all bases. In theory, if we find "the" solution for any base, we've found it for all of them, because we have an exact base conversion formula.
Edit: It really bugs me that all the tags in TeX start with backslashes, but the closing [/tex] tag starts with a forward slash.
