Consider a random sample of size n from a distribution with pdf

$f(x;q)=(Ln(q))^x/qx!$ if $x=0,1,...$ and $q>1$and $0$ otherwise.

a)Find a complete sufficient statistic for $q$

b)Find the maximum likelihood estimator of $q$

c)Find the Cramer-Rao Lower Bound for $q$

d)Find the Uniformly Minimum Variance Unbiased Estimator for $Ln(q)$

e)Find the Uniformly Minimum Variance Unbiased Estimator for $(Ln(q))^2$

(f) Find the Cramer-Rao Lower Bound for $(Ln(q))^2$

for part a) can be writen as $1/qx!e^{xLn(Ln(q))}$which shows that this is regular exponent. and by Theorem, we know $S=\Sigma X$ is complete suff stat. But how do we find the MLE of q?
for the part b) Find the maximum likelihood estimator of . i found that MLE of $q=e^{\overline{X}}$ where $\overline{X}=\Sigma X/n$. and now i am trying to do part c)Find the Cramer-Rao Lower Bound for . A theorem tells me that (i hope this is correct) to find CRLB of $q$, find $Var(e^{\overline{X}})$.