Existence, Uniqueness & Bifurcations · Challenge problem

Near $r = 1$ the family $y' = r\ln(1 + y) - y$ (for $y > -1$) has a transcritical bifurcation at $y = 0$: besides $y = 0$ there is an equilibrium $y^*(r)$ with $y^*(1) = 0$. Find $\displaystyle\lim_{r\to 1}\dfrac{y^*(r)}{r - 1}$.