Why is there a logarithm in the entropy formula? Why is it S=kln(N) where k is the Boltzmann constant and N is the number of microstates? Why isn't it S=N?
The reason that I know of, is that we require entropy to be an extensive property.
Suppose that we have two systems, with N1 and N2 microstates, respectively, and we join them. From basic statistics it follows that the new system has N = N1N2 microstates.
However, to be an extensive quantity, the entropy should scale as
S = S1 + S2.