  Introducing a suitable solution concept, we show that in bounded smooth
domains $\Omega\subset \mathbb{R}^n$, $n\ge 1$, the initial boundary value
problem for the chemotaxis system \begin{align*}
  u_t&=\Delta u -\chi\nabla\cdot\left(\frac{u}{v}\nabla v\right)+\kappa u -\mu
u^2,\\ v_t&=\Delta v -uv, \end{align*} with homogeneous Neumann boundary
conditions and widely arbitrary initial data has a generalized global solution
for any $\mu, \kappa, \chi >0$.
