Why I am getting similar $\beta$ for minimization $\sum_i (\log(y_i)-X_i\beta)^2 $ and $\sum_i (y_iX_i\beta-e^{X_i\beta})$?

Question

I asked a question here and learned that minimizing $\Vert \! \log(y)-X\beta \Vert_2^2$ and $\Vert y-e^{X\beta} \Vert_2^2$ are different.

But I still have difficult times to understand why minimization $\sum_i (\log(y_i)-X_i\beta)^2 $ and $\sum_i (y_iX_i\beta-e^{X_i\beta})$ will get "similar" results on $\beta$.

Could anyone show me why?