After reading about the new wikipedia page for the superlogarithm, I started thinking about my linear approximation for tetration.

I suppose I ought to write up something more formal. My linear approximation is the same as the "standard" linear approximation for base e, but otherwise it's different. And it has a couple big advantages over the "standard" linear approximation.

First of all, it's \( C^1 \) continuous, meaning it's continuous and once differentiable for all real x. The "standard" linear approximation is only \( C^0 \) continuous for bases other than e.

More importantly, it helps highlight the fact that there would be an inflection point on the critical interval that I define, which helps expose more information about the slog (and tetration). I've always been partial to simple formulae that expose additional insights.

Anyway, I first discussed my linear approximation on Google groups, which is where I first met Gottfried Helms incidentally. (Which reminds me, Gottfried had put forth a guess as to where the inflection point would be located. I still haven't tried to determine the precise location of the infleciton point and whether it's consistent from base to base.)

So I went back and looked more closely at this post:

http://groups.google.com/group/sci.math....d6e18a6185

I had been focussing on tetration at the time, but we can easily make it work for the slog as well, at least for real bases greater than eta (greater than 1 with a caveat) and using a real domain.

For a given base \( b \), let's find an integer constant \( n \), defined by:

\( \log_b^{\circ 2}(e) \lt \exp_b^{\circ n}(1) \le \log_b(e) \)

Having found this constant, we can now define the \( \mathrm{slog}_b(z) \) as:

\( \mathrm{slog}_b(z) = \begin{cases}

\mathrm{slog}_b(b^z) - 1 & \text{if } z \le \log_b^{\circ 2}(e) \\

\\[2pt]

\\

n + \frac{z-\exp_b^{\circ n}(1)}{\log_b(e)-\log_b^{\circ 2}(e)} & \text{if } \log_b^{\circ 2}(e) < z \le \log_b(e) \\

\\[5pt]

\\

\mathrm{slog}_b(\log_b(z)) + 1 & \text{if } \log_b(e) < z

\end{cases} \)

By the way, the caveat for bases between 1 and eta is that it gives us a linear approximation on the "corridor" between the upper and lower real fixed points. We would need to use complex numbers to generalize this outside this real interval, and my formula explicitly relies on use of real numbers. Therefore, rather than using 1 as a reference point for slog(z)=0, we would need to use a real number in the corridor. Using e is the simplest choice, though without looking at the complex slog, it's impossible to choose a "correct" reference point that corresponds with 1.

I suppose I ought to write up something more formal. My linear approximation is the same as the "standard" linear approximation for base e, but otherwise it's different. And it has a couple big advantages over the "standard" linear approximation.

First of all, it's \( C^1 \) continuous, meaning it's continuous and once differentiable for all real x. The "standard" linear approximation is only \( C^0 \) continuous for bases other than e.

More importantly, it helps highlight the fact that there would be an inflection point on the critical interval that I define, which helps expose more information about the slog (and tetration). I've always been partial to simple formulae that expose additional insights.

Anyway, I first discussed my linear approximation on Google groups, which is where I first met Gottfried Helms incidentally. (Which reminds me, Gottfried had put forth a guess as to where the inflection point would be located. I still haven't tried to determine the precise location of the infleciton point and whether it's consistent from base to base.)

So I went back and looked more closely at this post:

http://groups.google.com/group/sci.math....d6e18a6185

I had been focussing on tetration at the time, but we can easily make it work for the slog as well, at least for real bases greater than eta (greater than 1 with a caveat) and using a real domain.

For a given base \( b \), let's find an integer constant \( n \), defined by:

\( \log_b^{\circ 2}(e) \lt \exp_b^{\circ n}(1) \le \log_b(e) \)

Having found this constant, we can now define the \( \mathrm{slog}_b(z) \) as:

\( \mathrm{slog}_b(z) = \begin{cases}

\mathrm{slog}_b(b^z) - 1 & \text{if } z \le \log_b^{\circ 2}(e) \\

\\[2pt]

\\

n + \frac{z-\exp_b^{\circ n}(1)}{\log_b(e)-\log_b^{\circ 2}(e)} & \text{if } \log_b^{\circ 2}(e) < z \le \log_b(e) \\

\\[5pt]

\\

\mathrm{slog}_b(\log_b(z)) + 1 & \text{if } \log_b(e) < z

\end{cases} \)

By the way, the caveat for bases between 1 and eta is that it gives us a linear approximation on the "corridor" between the upper and lower real fixed points. We would need to use complex numbers to generalize this outside this real interval, and my formula explicitly relies on use of real numbers. Therefore, rather than using 1 as a reference point for slog(z)=0, we would need to use a real number in the corridor. Using e is the simplest choice, though without looking at the complex slog, it's impossible to choose a "correct" reference point that corresponds with 1.

~ Jay Daniel Fox