  We study a nonlinear Schr\"odinger equation with logarithmic nonlinearity on
a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a
boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We
investigate orbital stability and spectral instability of the standing wave
solutions $e^{i\omega t}\mathbf{\Phi}(x)$ to the equation when the profile
$\mathbf\Phi(x)$ has mixed structure (i.e. has bumps and tails). In our
approach we essentially use the extension theory of symmetric operators by
Krein - von Neumann, and the analytic perturbations theory.
