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

On the standing waves of the NLS-log equation with point interaction on a star graph