Editing Logarithm (section)

===Inverses of other exponential functions===
Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the [[logarithm of a matrix]] is the (multi-valued) inverse function of the [[matrix exponential]].<ref>{{Citation|last1=Higham|first1=Nicholas|author1-link=Nicholas Higham|title=Functions of Matrices. Theory and Computation|location=Philadelphia, PA|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-646-7|year=2008}}, chapter 11.</ref> Another example is the [[p-adic logarithm function|''p''-adic logarithm]], the inverse function of the [[p-adic exponential function|''p''-adic exponential]]. Both are defined via Taylor series analogous to the real case.<ref>{{Neukirch ANT}}, section II.5.</ref> In the context of [[differential geometry]], the [[exponential map (Riemannian geometry)|exponential map]] maps the [[tangent space]] at a point of a [[differentiable manifold|manifold]] to a [[neighborhood (mathematics)|neighborhood]] of that point. Its inverse is also called the logarithmic (or log) map.<ref>{{Citation|last1=Hancock|first1=Edwin R.|last2=Martin|first2=Ralph R.|last3=Sabin|first3=Malcolm A.|title=Mathematics of Surfaces XIII: 13th IMA International Conference York, UK, September 7–9, 2009 Proceedings|url=https://books.google.com/books?id=0cqCy9x7V_QC&pg=PA379|publisher=Springer|year=2009|page=379|isbn=978-3-642-03595-1}}</ref>

In the context of [[finite group]]s exponentiation is given by repeatedly multiplying one group element&nbsp;{{mvar|b}} with itself. The [[discrete logarithm]] is the integer&nbsp;''{{mvar|n}}'' solving the equation
<math display="block">b^n = x,</math>
where {{mvar|x}} is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in [[public key cryptography]], such as for example in the [[Diffie–Hellman key exchange]], a routine that allows secure exchanges of [[cryptography|cryptographic]] keys over unsecured information channels.<ref>{{Citation|last1=Stinson|first1=Douglas Robert|title=Cryptography: Theory and Practice|publisher=[[CRC Press]]|location=London|edition=3rd|isbn=978-1-58488-508-5|year=2006}}</ref> [[Zech's logarithm]] is related to the discrete logarithm in the multiplicative group of non-zero elements of a [[finite field]].<ref>{{Citation|last1=Lidl|first1=Rudolf|last2=Niederreiter|first2=Harald|author2-link=Harald Niederreiter|title=Finite fields|publisher=Cambridge University Press|isbn=978-0-521-39231-0|year=1997|url-access=registration|url=https://archive.org/details/finitefields0000lidl_a8r3}}</ref>

{{anchor|double logarithm}}Further logarithm-like inverse functions include the ''double logarithm''&nbsp;{{math|ln(ln(''x''))}}, the ''[[super-logarithm|super- or hyper-4-logarithm]]'' (a slight variation of which is called [[iterated logarithm]] in computer science), the [[Lambert W function]], and the [[logit]]. They are the inverse functions of the [[double exponential function]], [[tetration]], of {{math|''f''(''w'') {{=}} ''we<sup>w</sup>''}},<ref>{{Citation | last1=Corless | first1=R. | last2=Gonnet | first2=G. | last3=Hare | first3=D. | last4=Jeffrey | first4=D. | last5=Knuth | first5=Donald | author5-link=Donald Knuth | title=On the Lambert ''W'' function | url=http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf | year=1996 | journal=Advances in Computational Mathematics | issn=1019-7168 | volume=5 | pages=329–59 | doi=10.1007/BF02124750 | s2cid=29028411 | access-date=13 February 2011 | archive-url=https://web.archive.org/web/20101214110615/http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdf | archive-date=14 December 2010 | url-status=dead}}</ref> and of the [[logistic function]], respectively.<ref>{{Citation | last1=Cherkassky | first1=Vladimir | last2=Cherkassky | first2=Vladimir S. | last3=Mulier | first3=Filip | title=Learning from data: concepts, theory, and methods | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley series on adaptive and learning systems for signal processing, communications, and control | isbn=978-0-471-68182-3 | year=2007}}, p.&nbsp;357</ref>