From the perspective of pure mathematics, 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. In complex analysis and algebraic geometry, differential forms of the form df/f are known as forms with logarithmic poles.
The polylogarithm is the function defined by
It is related to the natural logarithm by Li1(z) = −ln(1 − z). Moreover, Lis(1) equals the Riemann zeta function ζ(s).
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