Given the probability density function $$f(x; \theta) = \frac{ \left(\ln(\theta)\right)^{x}}{\theta x!}, \quad x = 0,1,\ldots ; \theta > 1$$ and $0$ otherwise, find the Cramer-Rao Lower Bound for $\theta$.
I found a complete sufficient statistic for $\theta$ to be $S = \sum_{i=1}^{n}x_{i}$ and the Max-Likelihood Estimator of $\theta$ to be $ \hat{\theta} = e^{\bar{X}}$ where $ \bar{X} = { \sum_{i=1}^{n}\frac{x_{i}}{n}}$. Also $$\frac{d}{d\theta} \ln \left(L(\theta)\right)= \frac{-n}{\theta} + \frac{\sum_{i=1}^n x_{i}}{ \theta \ln(\theta)}$$
How do I find the CRLB for $\theta$ from here?