The log comes from the derivation of a function H satisfying certain natural requirements. See pg. 3 Sec. 2 of this source:

http://www.lptl.jussieu.fr/user/lesne/MSCS-entropy.pdf

Given the axioms, if you carry out the optimization, you get a unique (upto constants) function with a log in it.

All of the above answers are correct, except that they interpret the log, but don't explain the source of it.

The log comes from the derivation of a function H satisfying certain natural requirements. See pg. 3 Sec. 2 of this source:

Lesne, A. (2014). Shannon entropy: a rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics. Mathematical Structures in Computer Science, 24(3), e240311. PDF.

Given the axioms, if you carry out the optimization, you get a unique (upto constants) function with a log in it.

All of the above answers are correct, except that they interpret the log, but don't explain the source of it.