$\operatorname{Cov} (\overline{\log(X)}, \log(\bar{X}))$?

Question

Given a sample $(X_1,\ldots,X_n)$ that are uncorrelated, how can I simplify this even more : $$\operatorname{Cov} (\overline{\log(X_i)}, \log(\bar{X}))= \frac 1 n \sum \operatorname{Cov}(\log(\bar{X}), \log(X_i))$$ where $\bar{X}$ is the average of the sample and $\overline{\log(X_i)}$ is the average of $\log X$.