section_path: Generalizations/Related concepts
section_name: Related concepts
paragraph_text: Related concepts
  
  From the perspective of group theory, the identity log(cd) = log(c) + log(d) expresses a group isomorphism between positive reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups. By means of that isomorphism, the Haar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure dx / x on the positive reals. 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 one-forms df / f appear in complex analysis and algebraic geometry as differential forms with logarithmic poles.
  
  The polylogarithm is the function defined by Li s (z) = ∑ k = 1 ∞ z k k s. {\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.} It is related to the natural logarithm by Li (z) = −ln(1 − z). Moreover, Li (1) equals the Riemann zeta function ζ(s).