= Solution
The <sample mean> has <expectation> $\theta$ and <variance> $\theta^2/n$. For $g(z)=\log z$, $g'(\theta)=1/\theta$. The first-order delta-method expansion is
$$
\log\overline Z\approx\log\theta+\frac{\overline Z-\theta}{\theta}.
$$
Taking its mean and <variance> gives
$$
\boxed{\mathbb E[\log\overline Z]\approx\log\theta,\qquad\operatorname{Var}(\log\overline Z)\approx\frac1n.}
$$
These are large-sample approximations. Retaining the quadratic Taylor term would add the leading mean correction $-1/(2n)$, but it does not change the requested first-order expression.
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