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+#REDIRECT [[Iterated logarithm]]
-{{Refimprove|date=November 2007}}
-In [[mathematics]], the '''super-logarithm''' (or '''tetra-logarithm'''<ref>http://oeis.org/wiki/Tetra-logarithms</ref>) is one of the two inverse functions of [[tetration]]. Just as [[exponentiation]] has two inverse functions, [[Nth root|roots]] and [[logarithms]], [[tetration]] has two inverse functions, [[super-root]]s and super-logarithms. There are several ways of interpreting super-logarithms:
-* As the [[Abel function]] of [[exponential function]]s,
-* As the inverse function of [[tetration]] with respect to the height,
-* As the number of times a [[logarithm]] must be [[iterated function|iterated]] to get to 1 (the [[Iterated logarithm]]),
-* As a generalization of Robert Munafo's [http://www.mrob.com/pub/math/largenum.html large number class system],
-The precise definition of the super-logarithm depends on a precise definition of non-integral [[tetration]] (that is, <math> {^{y} x} </math> for ''y'' not an integer). There is no clear consensus on the definition of non-integral [[tetration]] and so there is likewise no clear consensus on the super-logarithm for non-integer [[range (mathematics)|range]].
-==Definitions==
-The super-logarithm, written <math>\operatorname{slog}_b(z),</math> is defined implicitly by
-:<math>\operatorname{slog}_b(b^z) = \operatorname{slog}_b(z) + 1</math> and
-:<math>\operatorname{slog}_b(1) = 0.</math>
-This definition implies that the super-logarithm can only have integer outputs, and that it is only defined for inputs of the form <math>b, b^b, b^{b^b},</math> and so on. In order to extend the domain of the super-logarithm from this sparse set to the real numbers, several approaches have been pursued. These usually include a third requirement in addition to those listed above, which vary from author to author. These approaches are as follows:
-* The linear approximation approach by Rubstov and Romerio,
-* The quadratic approximation approach by Andrew Robbins,
-* The regular Abel function approach by George Szekeres,
-* The iterative functional approach by Peter Walker, and
-* The natural matrix approach by Peter Walker, and later generalized by Andrew Robbins.
-== Approximations ==
-Usually, the [[special functions]] are defined not only for the real values of argument(s), but to complex plane, and differential and/or integral representation, as well as expansions in convergent and asymptotic series. Yet, no such representations are available for the '''slog''' function. Nevertheless, the simple approximations below are suggested.
-===Linear approximation===
-The linear approximation to the super-logarithm is:
-:<math>\operatorname{slog}_b(z) \approx \begin{cases}
-\operatorname{slog}_b(b^z) - 1 & \text{if } z \le 0 \\
--1 + z & \text{if } 0 < z \le 1 \\
-\operatorname{slog}_b(\log_b(z)) + 1 & \text{if } 1 < z \\
-\end{cases}</math>
-which is a piecewise-defined function with a linear "critical piece". This function has the property that it is continuous for all real ''z'' (<math>C^0</math> continuous). The first authors to recognize this approximation were Rubstov and Romerio, although it is not in [http://forum.wolframscience.com/showthread.php?s=&threadid=579 their paper], it can be found in [http://forum.wolframscience.com/showthread.php?threadid=956 their algorithm] that is used in their software prototype. The linear approximation to [[tetration]], on the other hand, had been known before, for example by [[Ioannis Galidakis]]. This is a natural inverse of the linear approximation to [[tetration]].
-Authors like Holmes recognize that the super-logarithm would be a great use to the next evolution of computer floating-point arithmetic, but for this purpose, the function need not be infinitely differentiable. Thus, for the purpose of representing large numbers, the linear approximation approach provides enough continuity (<math>C^0</math> continuity) to ensure that all real numbers can be represented on a super-logarithmic scale.
-===Quadratic approximation===
-The [[quadratic approximation]] to the super-logarithm is:
-:<math>\operatorname{slog}_b(z) \approx \begin{cases}
-\operatorname{slog}_b(b^z) - 1 & \text{if } z \le 0 \\
--1 + \frac{2\log(b)}{1+\log(b)}z +
-\frac{1-\log(b)}{1+\log(b)}z^2 & \text{if } 0 < z \le 1 \\
-\operatorname{slog}_b(\log_b(z)) + 1 & \text{if } 1 < z
-\end{cases}</math>
-which is a piecewise-defined function with a quadratic "critical piece". This function has the property that it is continuous and differentiable for all real ''z'' (<math>C^1</math> continuous). The first author to publish this approximation was Andrew Robbins in [https://web.archive.org/web/20090201164836/http://tetration.itgo.com/paper.html this paper].
-This version of the super-logarithm allows for basic calculus operations to be performed on the super-logarithm, without requiring a large amount of solving beforehand. Using this method, basic investigation of the properties of the super-logarithm and [[tetration]] can be performed with a small amount of computational overhead.
-== Approaches to the Abel function ==
-{{Main article|Abel function}}
-The Abel function is any function that satisfies Abel's functional equation:
-:<math>A_f(f(x)) = A_f(x) + 1</math>
-Given an Abel function <math>A_{f}(x)</math> another solution can be obtained by adding any constant <math>A'_{f}(x) = A_{f}(x) + c</math>. Thus given that the super-logarithm is defined by <math>\operatorname{slog}_b(1) = 0</math> and the third special property that differs between approaches, the Abel function of the exponential function could be uniquely determined.
-==Properties==
-Other equations that the super-logarithm satisfies are:
-:<math>\operatorname{slog}_b(z) = \operatorname{slog}_b(\log_b(z)) + 1</math>
-:<math>\operatorname{slog}_b(z) > -2</math> for all real ''z''
-Probably the first example of mathematical problem where the solution is expressed in terms of super-logarithms, is the following:
-: Consider oriented graphs with ''N'' nodes and such that oriented path from node ''i'' to node ''j'' exists if and only if <math>i>j.</math> If length of all such paths is at most ''k'' edges, then the minimum possible total number of edges is:
-:: <math>\Theta(N^2)</math> for <math>k=1</math>
-:: <math>\Theta(N \log N)</math> for <math>k=2</math>
-:: <math>\Theta(N \log \log N)</math> for <math>k=3</math>
-:: <math>\Theta(N \operatorname{slog} N)</math> for <math>k=4</math> and <math>k=5</math>
-:(M. I. Grinchuk, 1986;<ref>М. И. Гринчук, ''О сложности реализации последовательности треугольных булевых матриц вентильными схемами различной глубины'', in: Методы дискретного анализа в синтезе управляющих систем, 44 (1986), pp. 3—23.</ref> cases <math>k>5</math> require super-super-logarithms, super-super-super-logarithms etc.)
-==Super-logarithm as inverse of tetration==
-[[Image:Slogez01.jpg|right|256px|thumb|<math>f={\rm slog}_{\rm e}(z)</math> in the complex z-plane.]]
-As [[tetration]] (or super-exponential) <math>~{\rm sexp}_b(z)~</math> is suspected to be an analytic function,<ref name="walker">{{cite journal
-|author=Peter Walker
-|title=Infinitely Differentiable Generalized Logarithmic and Exponential Functions
-|journal=[[Mathematics of Computation]]
-|volume=57
-|issue=196
-|year=1991
-|pages=723–733
-|doi=10.2307/2938713
-|jstor=2938713
-|publisher=American Mathematical Society
-}}
-</ref> at least for some values of <math>~b~</math>, the inverse function <math> {\rm slog}_b = {\rm sexp}_b^{-1}</math> may also be analytic.
-Behavior of
-<math>~{\rm slog}_b(z)~</math>, defined in such a way, the complex <math>~z~</math> plane is sketched in Figure 1 for the case <math>~b=e~</math>. Levels of integer values of real and integer values of imaginary parts of the slog functions are shown with thick lines.
-If the existence and uniqueness of the [[analytic extension]] of [[tetration]] is provided by the condition of its asymptotic approach to the [[Fixed point (mathematics)|fixed points]]
-<math> L \approx 0.318 + 1.337{\!~\rm i} </math> and
-<math> L^* \approx 0.318 - 1.337{\!~\rm i} </math>
-of <math> L=\ln(L) </math><ref name="hellmuth50">
-{{cite journal
-|journal=[[Journal für die reine und angewandte Mathematik]]
-|author=H.Kneser
-|title=Reelle analytische Losungen der Gleichung  <math>\varphi\Big(\varphi(x)\Big)={\rm e}^x</math>  und verwandter Funktionalgleichungen
-|volume=187
-|pages=56–67
-|year=1950
-}}</ref>
-in the upper and lower parts of the complex plane, then the inverse function should also be unique.
-Such a function is real at the real axis. It has two [[branch point]]s at
-<math>~z=L~</math>  and
-<math>~z=L^*</math>. It approaches its limiting value <math>-2</math> in vicinity of the negative part of the real axis (all the strip between the cuts shown with pink lines in the figure), and slowly grows up along the positive
-direction of the real axis.
-As the derivative at the real axis is positive, the imaginary part of slog remains positive just above the real axis and negative just below the real axis.
-The existence, uniqueness and generalizations are under discussion.<ref name="forum">Tetration forum, http://math.eretrandre.org/tetrationforum/index.php</ref>
-==See also==
-* [[Iterated logarithm]]
-* [[Tetration]]
-==References==
-<references/>
-* Ioannis Galidakis, [http://ioannis.virtualcomposer2000.com/math/ Mathematics], published online (accessed Nov 2007).
-* W. Neville Holmes, [http://portal.acm.org/citation.cfm?id=620661 Composite Arithmetic: Proposal for a New Standard], IEEE Computer Society Press, vol. 30, no. 3, pp.&nbsp;65–73, 1997.
-* Robert Munafo, [http://www.mrob.com/pub/math/largenum.html Large Numbers at MROB], published online (accessed Nov 2007).
-* C. A. Rubtsov and G. F. Romerio, [http://forum.wolframscience.com/showthread.php?s=&threadid=579 Ackermann's Function and New Arithmetical Operation], published online (accessed Nov 2007).
-* Andrew Robbins, [http://tetration.itgo.com/paper.html Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm], published online (accessed Nov 2007).
-* George Szekeres, [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.em/1048515657   Abel's equation and regular growth]: variations on a theme by Abel, Experiment. Math. Volume 7, Issue 2 (1998), 85-100.
-* Peter Walker, [https://www.jstor.org/stable/2938713 Infinitely Differentiable Generalized Logarithmic and Exponential Functions], Mathematics of Computation, Vol. 57, No. 196 (Oct., 1991), pp.&nbsp;723–733.
-==External links==
-* Rubstov and Romerio, [http://forum.wolframscience.com/showthread.php?threadid=956 Hyper-operations Thread 1]
-* Rubstov and Romerio, [http://forum.wolframscience.com/showthread.php?threadid=579 Hyper-operations Thread 2]
-{{Hyperoperations}}
-{{DEFAULTSORT:Super-Logarithm}}
-[[Category:Logarithms]]
-[[tr:Tetrasyon#Süper logaritma|Süper logaritma]]