We prove the statement$/$conjecture of M. Kontsevich on the existence of the logarithmic formality morphism $\mathcal{U}^{log}$. This question was open since 1999, and the main obstacle was the presence of $dr/r$ type singularities near the boundary $r = 0$ in the integrals over compactified configuration spaces. The novelty of our approach is the use of local torus actions on configuration spaces of points in the upper half-plane. It gives rise to a version of Stokes' formula for differential forms with singularities at the boundary which implies the formality property of $\mathcal{U}^{log}$. We also show that the logarithmic formality morphism admits a globalization from $\mathbb{R}^{d}$ to an arbitrary smooth manifold.[-]