About Super-logarithms

A super-logarithm is an inverse function of tetration, specifically the inverse of a tetrational function. Some have called super-logarithms anti-tetrational functions [?] because of this relationship. There are several interpretations of super-logarithms that stem from this definition:

slog_x(`z`) = `y` if and only if `z` = ^y`x`	A super-logarithm is an inverse of a tetrational function.
slog_x(`z`) = `A`_z[`x`^z](`z`)	A super-logarithm is the Abel function of an exponential function.
log^[slog(z)](`z`) = 1	A super-logarithm represents how many times a logarithm must be applied to a number before a result of one is obtained.
`C`(`n`) = ceiling( slog₁₀(`n`) - slog₁₀(6) )	Super-logarithms are related to Munafo's number class system.

Integer Super-logarithms

Munafo's number class system

In Munafo's website about Large Numbers, he describes a number class system. Munafo's number class system begins with defining class-0 numbers as 1 through 6. He then gives an upper-bound for class-1 at about 10⁶, and for class-2 numbers at about 10^(10^6). This is a super-logarithmic scale, so we can use the super-logarithm to simplify this description.

Let the Munafo number of n be denoted C(n). All numbers in the class C(n) satisfy: \begin{equation} \exp^{C(n) - 1}_{10}(6) < n \le \exp^{C(n)}_{10}(6) \end{equation} Replace 6 with $\exp^{\slog_{10}(6)}_{10}(1)$, and add hyper-exponents: \begin{align} \exp^{C(n) - 1}_{10}{\left(\exp^{\slog_{10}(6)}_{10}(1)\right)} & < n \le \exp^{C(n)}_{10}{\left(\exp^{\slog_{10}(6)}_{10}(1)\right)} \\ \exp^{C(n) - 1 + \slog_{10}(6)}_{10}(1) & < n \le \exp^{C(n) + \slog_{10}(6)}_{10}(1) \end{align} Take the super-logarithm of each side, then subtract $C(n) + \slog_{10}(6)$ from each side: \begin{align} C(n) - 1 + \slog_{10}(6) < \slog_{10}(n) \le C(n) &{} + \slog_{10}(6) \\ - 1 < \slog_{10}(n) - \slog_{10}(6) - C(n) &{} \le 0 \\ \lceil \slog_{10}(n) - \slog_{10}(6) - C(n) &{} \rceil = 0 \end{align}

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