  We derive the asymptotic decay of the unique positive, radially symmetric
solution to the logarithmic Choquard equation $$ - \Delta u + a u = \frac{1}{2
\pi} \Bigl[\ln \frac{1}{|x|}* |u|^2 \Bigr] \ u \qquad \text{in $\mathbb{R}^2$}
$$ and we establish its nondegeneracy. For the corresponding three-dimensional
problem, the nondegeneracy property of the positive ground state to the
Choquard equation was proved by E. Lenzmann (Analysis & PDE, 2009).
