I'm studying quantum microcanonical ensemble, in particular an ideal system of $N$ particles with angular momentum $j=1/2$ which interact with an external magnetic field $B$. The entrophy $S$ of the system is

$$S=Nk_B\left[\left(\frac{E}{2N\mu B}-\frac{1}{2}\right)\ln\left(\frac{1}{2}-\frac{E}{2N\mu B}\right)-\left(\frac{E}{2N\mu B}+\frac{1}{2}\right)\ln\left(\frac{1}{2}+\frac{E}{2N\mu B}\right)\right].$$

It was obtained by

$$S = k_B \ln \omega(E;N;B)$$

Then, I studied that this expression of the entrophy has a maximum for $E=0$ and the minimums are locatted in $E=-\mu B N$ and $E=-\mu B N$. I also found the equation for the temperature

$$T=\frac{2\mu B}{k_B}\frac{1}{\ln\left(\frac{NB\mu-E}{NB\mu +E}\right)},$$

which is unbounded in $\mathbb{R}$ like you can see easily. 

However, I don't know if I the definition of the entrophy

$$S = k_B \ln \Phi(E;N;B)$$

is equivalent to the other. 

I look for the answer to this question and I find it in a book. It says that they are different because in $E\geq 0$ the number of quantum states don't grow when the energy grows up. Despite I have this answer, I don't understand it very much. 

What does it mean with it?

Notation: 

$\omega(E,N,B)$ denoe the number of leveles with energy $E-\frac{\Delta E}{2}<E_n<E+\frac{\Delta E}{2}$ 

$\Phi(E,N,B)$ denote $\int \omega(E,N,B)dE$