  In this paper, we study constraint minimizers $u$ of the planar
Schr\"odinger-Poisson system with a logarithmic convolution potential $\ln
|x|\ast u^2$ and a logarithmic external potential $V(x)=\ln (1+|x|^2)$, which
can be described by the $L^2$-critical constraint minimization problem with a
subcritical perturbation. We prove that there is a threshold $\rho ^* \in
(0,\infty)$ such that constraint minimizers exist if and only if
$0<\rho<\rho^*$. In particular, the local uniqueness of positive constraint
minimizers as $\rho\nearrow\rho^*$ is analyzed by overcoming the sign-changing
property of the logarithmic convolution potential and the non-invariance under
translations of the logarithmic external potential.
