Differential Equations: JEE Main 2023 Mathematics question

If the solution curve $f(x, y)=0$ of the differential equation $\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0$, passes through the points $(1,0)$ and $(\alpha, 2)$, then $\alpha^{\alpha}$ is equal to :