In this paper, for every $$\alpha \in \mathbb {R}$$ , we characterize $$C_{\mathcal {B}_{\log }}(\mathcal {D}_\alpha \cap \mathcal {B}_{\log })$$ , the closure of Dirichlet type space $$\mathcal {D}_\alpha $$ in the logarithmic Bloch space $$\mathcal {B}_{\log }$$ . For the case of $$\alpha =0$$ , we answer a question raised by Qian and Li recently. We also consider the strict inclusion relation among the little logarithmic Bloch space, $$C_{\mathcal {B}_{\log }}(\mathcal {D}_\alpha \cap \mathcal {B}_{\log })$$ and $$\mathcal {B}_{\log }$$ . In addition, we revisit a description of the boundedness of composition operator from $$\mathcal {B}_{\log }$$ to $$C_{\mathcal {B}_{\log }}(\mathcal {D}_\alpha \cap \mathcal {B}_{\log })$$ .