Why Boltzmann Entropy's formula is $k_B\log\ W$ and not $0.5\ k_B\log\ W$? This question is probably wrong - however, I wish to understand even why it's wrong..
Let's take the monatomic gas situation. By the Equipartition theorem, we have that for every degree of freedom the mean energy is $0.5\ k_BT$. So, why in the definition of the Boltzmann Entropy we have $k_B\log\ W$ and not $0.5\ k_B\log\ W$?
 A: The problem that I see with the idea is that it seems contradictory. You used the definition of entropy of the microcanonical ensemble, which is defined by assigning an equal probability to every microstate whose energy falls within a range centered at E. All other microstates are given a probability of zero. That is,  the range of energy is reduced in width until it is infinitesimally narrow, still centered at E( In the limit of this process, the microcanonical ensemble is obtained).
The equipartition theorem , on the other hand, allows a range of energies of which $\frac{1}{2}kT$ is the average. It is not surprising then that the mixing of the two formalisms result in
 (I would not say contradictory) non consistent results.
A: Assume we already know the relation $S\propto\log{W}$. Then we can define a coefficient of proportionality as $k_\text{B}$ and write $S=k_\text{B}\log{W}$. So it is defined including all possible multiplication ambiguities from the beginning. Basically, this is a same question asking why we should choose $m$ not $2m$ or something else in Newton's second law, $F=ma$.
