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In this paper we study the one-dimensional logarithmic Schr\"odinger equationperturbed by an attractive $\delta^{\prime}$-interaction \[i\partial_{t}u+\partial^{2}_{x}u+ \gamma\delta^{\prime}(x)u+u\,\mbox{Log}\left|u\right|^{2}=0, \quad (x,t)\in\mathbb{R}\times\mathbb{R}, \]where $\gamma>0$. We establish the existence and uniqueness of the solutions ofthe associated Cauchy problem in a suitable functional framework. In theattractive $\delta^{\prime}$-interaction case, the set of the ground state iscompletely determined. More precisely: if $0<\gamma\leq 2$, then there is asingle ground state and it is an odd function; if $\gamma>2$, then there existtwo non-symmetric ground states. Finally, we show that the ground states areorbitally stable via a variational approach.

Stability of ground states for logarithmic Schr\"{o}dinger equation with a $\delta^{\prime}$-interaction