Editing Logarithm (section)

==Generalizations==

===Complex logarithm===
{{Main|Complex logarithm}}
[[File:Complex number illustration multiple arguments.svg|thumb|right|Polar form of {{math|''z {{=}} x + iy''}}. Both {{mvar|φ}} and {{mvar|φ'}} are arguments of {{mvar|z}}.|alt=An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x-axis.]]

All the [[complex number]]s {{mvar|a}} that solve the equation

<math display="block">e^a=z</math>

are called ''complex logarithms'' of {{mvar|z}}, when {{mvar|z}} is (considered as) a complex number. A complex number is commonly represented as {{math|''z {{=}} x + iy''}}, where {{mvar|x}} and {{mvar|y}} are real numbers and {{mvar|i}} is an [[imaginary unit]], the square of which is −1. Such a number can be visualized by a point in the [[complex plane]], as shown at the right. The [[polar form]] encodes a non-zero complex number&nbsp;{{mvar|z}} by its [[absolute value]], that is, the (positive, real) distance&nbsp;{{Mvar|r}} to the [[origin (mathematics)|origin]], and an angle between the real ({{mvar|x}}) axis''&nbsp;''{{Math|Re}} and the line passing through both the origin and {{mvar|z}}. This angle is called the [[Argument (complex analysis)|argument]] of {{mvar|z}}.

The absolute value {{mvar|r}} of {{mvar|z}} is given by

<math display="block">\textstyle r=\sqrt{x^2+y^2}.</math>

Using the geometrical interpretation of [[sine]] and [[cosine]] and their periodicity in {{Math|2{{pi}}}}, any complex number&nbsp;{{mvar|z}} may be denoted as

<math display="block">\begin{align}
z &= x + iy \\
  &= r (\cos \varphi + i \sin \varphi ) \\
  &= r (\cos (\varphi + 2k\pi) + i \sin (\varphi + 2k\pi)),
\end{align}</math>

for any integer number&nbsp;{{mvar|k}}. Evidently the argument of {{mvar|z}} is not uniquely specified: both {{mvar|φ}} and {{Math|1=''φ''' = ''φ'' + 2''k''{{pi}}}} are valid arguments of {{mvar|z}} for all integers&nbsp;{{mvar|k}}, because adding {{Math|2''k''{{pi}}}}&nbsp;[[radian]]s or ''k''⋅360°{{refn|See [[radian]] for the conversion between 2[[pi|{{pi}}]] and 360 [[degree (angle)|degree]].|group=nb}} to {{mvar|φ}} corresponds to "winding" around the origin counter-clock-wise by {{mvar|k}}&nbsp;[[Turn (geometry)|turns]]. The resulting complex number is always {{mvar|z}}, as illustrated at the right for {{math|''k'' {{=}} 1}}. One may select exactly one of the possible arguments of {{mvar|z}} as the so-called ''principal argument'', denoted {{math|Arg(''z'')}}, with a capital&nbsp;{{math|A}}, by requiring {{mvar|φ}} to belong to one, conveniently selected turn, e.g. {{Math|−{{pi}} < ''φ'' ≤ {{pi}}}}<ref>{{Citation|last1=Ganguly|location=Kolkata|first1=S.|title=Elements of Complex Analysis|publisher=Academic Publishers|isbn=978-81-87504-86-3|year=2005}}, Definition 1.6.3</ref> or {{Math|0 ≤ ''φ'' < 2{{pi}}}}.<ref>{{Citation|last1=Nevanlinna|first1=Rolf Herman|author1-link=Rolf Nevanlinna|last2=Paatero|first2=Veikko|title=Introduction to complex analysis|journal=London: Hilger|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4399-4|year=2007|bibcode=1974aitc.book.....W}}, section 5.9</ref> These regions, where the argument of {{mvar|z}} is uniquely determined are called [[principal branch|''branches'']] of the argument function.

[[File:Complex log domain.svg|right|thumb|The principal branch (-{{pi}}, {{pi}}) of the complex logarithm, {{math|Log(''z'')}}. The black point at {{math|''z'' {{=}} 1}} corresponds to absolute value zero and brighter colors refer to bigger absolute values. The [[hue]] of the color encodes the argument of {{math|Log(''z'')}}.|alt=A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.]]

[[Euler's formula]] connects the [[trigonometric functions]] [[sine]] and [[cosine]] to the [[complex exponential]]:
<math display="block">e^{i\varphi} = \cos \varphi + i\sin \varphi .</math>

Using this formula, and again the periodicity, the following identities hold:<ref>{{Citation|last1=Moore|first1=Theral Orvis|last2=Hadlock|first2=Edwin H.|title=Complex analysis|publisher=[[World Scientific]]|location=Singapore|isbn=978-981-02-0246-0|year=1991}}, section 1.2</ref>

<math display="block"> \begin{align}
z &= r \left (\cos \varphi + i \sin \varphi\right) \\
  &= r \left (\cos(\varphi + 2k\pi) + i \sin(\varphi + 2k\pi)\right) \\
  &= r e^{i (\varphi + 2k\pi)} \\
  &= e^{\ln(r)} e^{i (\varphi + 2k\pi)}  \\
  &= e^{\ln(r) + i(\varphi + 2k\pi)} = e^{a_k},
\end{align}
</math>

where {{math|ln(''r'')}} is the unique real natural logarithm, {{math|''a''<sub>''k''</sub>}} denote the complex logarithms of {{mvar|z}}, and {{mvar|k}} is an arbitrary integer. Therefore, the complex logarithms of {{mvar|z}}, which are all those complex values {{math|''a''<sub>''k''</sub>}} for which the {{math|''a''<sub>''k''</sub>-th}}&nbsp;power of {{mvar|e}} equals {{mvar|z}}, are the infinitely many values
<math display="block">a_k = \ln (r) + i ( \varphi + 2 k \pi ),</math>
for arbitrary integers {{mvar|k}}.

Taking {{mvar|k}} such that {{Math|''φ'' + 2''k''{{pi}}}} is within the defined interval for the principal arguments, then {{math|''a''<sub>''k''</sub>}} is called the ''principal value'' of the logarithm, denoted {{math|Log(''z'')}}, again with a capital&nbsp;{{math|L}}. The principal argument of any positive real number&nbsp;{{mvar|x}} is 0; hence {{math|Log(''x'')}} is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers [[Exponentiation#Failure of power and logarithm identities|do {{em|not}} generalize]] to the principal value of the complex logarithm.<ref>{{Citation | last1=Wilde | first1=Ivan Francis | title=Lecture notes on complex analysis | publisher=Imperial College Press | location=London | isbn=978-1-86094-642-4 | year=2006|url=https://books.google.com/books?id=vrWES2W6vG0C&q=complex+logarithm&pg=PA97}}, theorem 6.1.</ref>

The illustration at the right depicts {{math|Log(''z'')}}, confining the arguments of {{mvar|z}} to the interval {{open-closed|−π, π}}. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real {{mvar|x}}&nbsp;axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding {{mvar|k}}-value of the continuously neighboring branch. Such a locus is called a [[branch cut]]. Dropping the range restrictions on the argument makes the relations "argument of {{mvar|z}}", and consequently the "logarithm of {{mvar|z}}", [[multi-valued function]]s.

===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>

===Related concepts===
From the perspective of [[group theory]], the identity {{math|log(''cd'') {{=}} log(''c'') + log(''d'')}} expresses a [[group isomorphism]] between positive [[real number|reals]] under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups.<ref>{{Citation|last1=Bourbaki|first1=Nicolas|author1-link=Nicolas Bourbaki|title=General topology. Chapters 5–10|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=Elements of Mathematics|isbn=978-3-540-64563-4|mr=1726872|year=1998}}, section V.4.1</ref> By means of that isomorphism, the [[Haar measure]] ([[Lebesgue measure]])&nbsp;{{math|''dx''}} on the reals corresponds to the Haar measure&nbsp;{{math|''dx''/''x''}} on the positive reals.<ref>{{Citation|last1=Ambartzumian|first1=R.V.|author-link=Rouben V. Ambartzumian|title=Factorization calculus and geometric probability|publisher=[[Cambridge University Press]]|isbn=978-0-521-34535-4|year=1990|url-access=registration|url=https://archive.org/details/factorizationcal0000amba}}, section 1.4</ref> The non-negative reals not only have a multiplication, but also have addition, and form a [[semiring]], called the [[probability semiring]]; this is in fact a [[semifield]]. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition ([[LogSumExp]]), giving an [[isomorphism]] of semirings between the probability semiring and the [[log semiring]].

[[logarithmic form|Logarithmic one-forms&nbsp;]]{{math|''df''/''f''}} appear in [[complex analysis]] and [[algebraic geometry]] as [[differential form]]s with logarithmic [[Pole (complex analysis)|poles]].<ref>{{Citation|last1=Esnault|first1=Hélène|last2=Viehweg|first2=Eckart|title=Lectures on vanishing theorems|location=Basel, Boston|publisher=Birkhäuser Verlag|series=DMV Seminar|isbn=978-3-7643-2822-1|mr=1193913|year=1992|volume=20|doi=10.1007/978-3-0348-8600-0|citeseerx=10.1.1.178.3227}}, section 2</ref>

The [[polylogarithm]] is the function defined by
<math display="block">
\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}.
</math>
It is related to the [[natural logarithm]] by {{math|1=Li<sub>1</sub>&thinsp;(''z'') = −ln(1 − ''z'')}}. Moreover, {{math|Li<sub>''s''</sub>&thinsp;(1)}} equals the [[Riemann zeta function]] {{math|ζ(''s'')}}.<ref>{{dlmf|id= 25.12|first= T.M.|last= Apostol}}</ref>