A unique solution to $\log_2\prod_{a=1}^{a=C} (3+1/{M_a} ) \in \mathbb{Z}, C \in \mathbb{Z}, M_a \rm{~is~odd}, M_i \ne M_j$

Question

$$\log_2\prod_{a=1}^{a=C} \left (3 + \frac{1}{M_a} \right) \in \mathbb{Z}, \qquad C \in \mathbb{Z}, \qquad M_a \rm{~is~odd}, \qquad M_i \ne M_j$$

I've been playing with this equation and I'm fairly confident the only solution to this is: $C=1, M_a=1$

Can it be proven this solution is unique? Am I missing something obvious?