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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>\text{slog}_b(z)</math> or <math>\text{log}^{4}_b(z)</math>, is defined implicitly by
-:<math>\,\mathrm{slog}_b(b^z) = \mathrm{slog}_b(z) + 1</math> and
-:<math>\,\mathrm{slog}_b(1) = 0.</math>
-Notice that this definition can only have integer outputs, and will only accept values that will produce integer outputs. The only numbers that this definition will accept are 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>\mathrm{slog}_b(z) \approx \begin{cases}
-\mathrm{slog}_b(b^z) - 1 & \text{if } z \le 0 \\
--1 + z & \text{if } 0 < z \le 1 \\
-\mathrm{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>\mathrm{slog}_b(z) \approx \begin{cases}
-\mathrm{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 \\
-\mathrm{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>\mathrm{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>\,\mathrm{slog}_b(z) = \mathrm{slog}_b(\log_b(z)) + 1</math>
-:<math>\,\mathrm{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 \,\mathrm{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]]