Theta Manipulation to show $N = \Theta(n/\log N)=\Theta(n/\log n)$
There is not any strange thing here. As:

$$\frac{n}{\log(n)} = \frac{n}{\log(N\log(N))} = \frac{n}{\log(N)+\log(\log(N))} = \Theta\left(\frac{n}{\log(N)}\right) = O\left(\frac{n}{\log(N)}\right)$$

Therefore, all of them are true.