Editing Logarithm (section)

===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>