Statistical Inference — MLE reducing to the sample geometric mean

Consider a random sample $\{8,4,\frac12,1\}$ from the distribution having most general form of the probability mass function $f(x,\theta)=\left(\frac{x}{\theta}\right)^{\theta A'(\theta)}\exp[A(\theta)+C(x)]$ where $A'(\theta)$ is the derivative of $A(\theta)$ with respect to $\theta$. The maximum likelihood estimator of $\theta$ is