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In this paper, we consider the anisotropic Lorentz space $L_{\bar{p}, \bar\theta}^{*}(\mathbb{I}^{m})$ of periodic functions of $m$ variables. The Besov space $B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}$ of functions with logarithmic smoothness is defined. The aim of the paper is to find an exact order of the best approximation of functions from the class $B_{\bar{p}, \bar\theta}^{(0, \alpha, \tau)}$ by trigonometric polynomials under different relations between the parameters $\bar{p}, \bar\theta,$ and $\tau$.The paper consists of an introduction and two sections. In the first section, we establish a sufficient condition for a function $f\in L_{\bar{p}, \bar\theta^{(1)}}^{*}(\mathbb{I}^{m})$ to belong to the space $L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})$ in the case $1{&lt;\theta^{2}&lt;\theta_{j}^{(1)}},$ ${j=1,\ldots,m},$ in terms of the best approximation and prove its unimprovability on the class $E_{\bar{p},\bar{\theta}}^{\lambda}=\{f\in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})\colon{E_{n}(f)_{\bar{p},\bar{\theta}}\leq\lambda_{n},}$ ${n=0,1,\ldots\},}$ where $E_{n}(f)_{\bar{p},\bar{\theta}}$ is the best approximation of the function $f \in L_{\bar{p},\bar{\theta}}^{*}(\mathbb{I}^{m})$ by trigonometric polynomials of order $n$ in each variable $x_{j},$ $j=1,\ldots,m,$ and $\lambda=\{\lambda_{n}\}$ is a sequence of positive numbers $\lambda_{n}\downarrow0$ as $n\to+\infty$. In the second section, we establish order-exact estimates for the best approximation of functions from the class $B_{\bar{p}, \bar\theta^{(1)}}^{(0, \alpha, \tau)}$ in the space $L_{\bar{p}, \theta^{(2)}}^{*}(\mathbb{I}^{m})$.

EESTIMATES OF BEST APPROXIMATIONS OF FUNCTIONS WITH LOGARITHMIC SMOOTHNESS IN THE LORENTZ SPACE WITH ANISOTROPIC NORM