  We study analytically the orbital stability of the standing waves with a
peak-Gausson profile for a nonlinear logarithmic Schr\"odinger equation with
$\delta$-interaction (attractive and repulsive). A major difficulty is to
compute the number of negative eigenvalues of the linearized operator around
the standing wave. This is overcome by the perturbation method, the
continuation arguments, and the theory of extensions of symmetric operators.
