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In nearly all areas of mathematics, logarithmic function applied to various objects where it is defined, produces either direction data or scale data of the object.

Are there any examples of mathematical objects that have logarithmic functions defined on them, that produce other characteristic of the object besides the direction and the scale?

I think of something exotic, like knots, singularities, polyhedrons, graphs, etc.

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    $\begingroup$ What do you mean by "logarithmic function applied to objects"? Traditional logarithms can be applied only to numbers or functions, and those can represent whatever data one wishes. For example, logarithms of partition functions defined on knots (they encode knot invariants) are free energies, and both come up in string theory. Do you mean things loosely inspired by logarithms like log structures and logarithmic functors? $\endgroup$ – Conifold Apr 2 at 20:35
  • $\begingroup$ One application direction data or scale data (number) is the log of matrix as opposite to the exponential of matrix. $\endgroup$ – hermes Apr 9 at 13:58

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