Let \( y=y(x) \) be the solution of the differential equal to \( dy=e^{\alpha x + y} \, dx:\ \alpha \in \mathbb{N} \). If \( y(\log_{e} 2)=\log_{e} 2 \) and \( y(0)=\log_{e}\!\left(\dfrac{1}{2}\right) \), then the value of \( \alpha \) is equal to ____________.
