  In this note, we characterize the equality case of the sharp $L^2$-Euclidean
logarithmic Sobolev inequality with monomial weights, exploiting the idea by
Bobkov and Ledoux \cite{Bob}. Our approach is new even in the unweighted case.
Also, we show that the same strategy yields the equality case of the sharp
$L^p$-Euclidean logarithmic Sobolev inequality for $1 < p < \infty$ with
arbitrary norm on ${\mathbb R}^n$, if the inequality is unweighted.
