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The article considers the Lorentz space $L_{p,\tau}(\mathbb{T}^{m})$, $2\pi$of periodic functions of many variables and $S_{p,\tau,\theta}^{0,\overline{b}}\mathbf{B}$, $S_{p, \tau, \theta}^{0, \overline{b}}B$ -- spaces offunctions with mixed logarithmic smoothness. The article establishes necessaryand sufficient conditions for embedding the spaces $S_{p, \tau, \theta}^{0,\overline{b}}\mathbf{B}$ and $S_{p, \tau, \theta}^{ 0, \overline{b}}B$ intoeach other.

On the exactness of the conditions of embedding theorems for spaces of functions with mixed logarithmic smoothness