Essential norm estimates for weighted composition operator on the logarithmic Bloch space

In this article, we estimate the essential norm of weighted composition operator $W_{u,\phi }$, acting on the logarithmic Bloch space ${\mathcal{B}}^{v_{\log }}$, in terms of the $n$-power of the analytic function $\phi$ and the norm of the $n$-power of the identity function. Also, we estimate the essential norm of the weighted composition operator from ${\mathcal{B}}^{v_{\log }}$ into the growth space $H_{v_{\log }}^{\mathrm{\infty }}$. As a consequence of our result, we estimate the essential norm of the composition operator $C_{\phi }$ acting on the Logarithmic-Zygmund space.