  We discuss the influence of possible spatial inhomogeneities in the
coefficients of logistic source terms in parabolic-elliptic chemotaxis-growth
systems of the form \begin{align*} u_t &= \Delta u - \nabla\cdot(u\nabla v) +
\kappa(x)u-\mu(x)u^2, 0 &= \Delta v - v + u \end{align*} in smoothly bounded
domains $\Omega\subset\mathbb{R}^2$. Assuming that the coefficient functions
satisfy $\kappa,\mu\in C^0(\overline{\Omega})$ with $\mu\geq0$ we prove that
finite-time blow-up of the classical solution can only occur in points where
$\mu$ is zero, i.e.\ that the blow-up set $\mathcal{B}$ is contained in
\begin{align*} \big\{x\in\overline{\Omega}\mid\mu(x)=0\big\}. \end{align*}
Moreover, we show that whenever $\mu(x_0)>0$ for some
$x_0\in\overline{\Omega}$, then one can find an open neighbourhood $U$ of $x_0$
in $\overline{\Omega}$ such that $u$ remains bounded in $U$ throughout
evolution.
