Super-logarithm

From Tetration.net - written by Andrew Robbins

The super-logarithm is one of the two inverses of tetration. The term super-logarithm probably derives from the term super-exponentiation which was an older name for tetration. The earliest use of the term super-logarithm is probably in the 1989 paper Ackermann's Function and New Arithmetical Operations by Rubstov and Romerio.

1 Values
2 Definitions
- 2.1 Piecewise definition
- 2.2 Series definition
  - 2.2.1 Copyright
3 References

Values

Some values of the super-logarithm for small values:
$slog x (0) = - 1$
$slog x (1) = 0$
$slog x (x) = 1$

Definitions

Almost all methods of defining the super-logarithm depend on the axiom:
$slog x (z) = slog x (x z) - 1$

Piecewise definition

We can piecewise-define the super-logarithm as:
$slog x (z) = s (x, x z) - 1$ if z ≤ 0
$slog x (z) = s (x, z)$ if 0 < z ≤ 1
$slog x (z) = s (x,log x (z))$ if z > 1

Series definition

We can also define a series that represents the super-logarithm as:
$s(x, z)_n = -1 + \sum_{k=1}^{n} z^k u_k(x)$ if x > 1
$s(x, z) = \lim_{n \rightarrow \infty} s(x, z)_n$ if x > 1
where $u k (x)$ satisfies the matrix equaion: ${\mathbf B} {\mathbf u} = {\mathbf b}$ where:
$B j k = k (j - 1) - δ (j - 1) k (k! / log(x) (j - 1))$
$u k = u k (x)$
$b j = δ (j - 1)0$
and $δ j k$ is Kronecker delta.

Copyright

The above formula for the matrix in the series definition of the super-logarithm is based on a copyrighted work by Andrew Robbins, who gives permission to use it on tetration.net.

References

C. A. Rubstov and G. F. Romerio, "Ackermann's Function and New Arithmetical Operations", 1989.

A. J. Robbins, "Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm", 2005.