UMVUE of $\theta$ when $X_1,\ldots,X_n$ are i.i.d with pdf $f(x)=\frac{(\ln\theta)\theta^x }{\theta -1}$

Question

I'm having some trouble finding the UMVU estimator of $\theta$ for the following distribution

$$f(x)=\frac{(\ln\theta)\theta^x }{\theta -1} \text{ for } x \in (0,1)$$

Specifically, I know $T_n=\sum x_i$ is a sufficient complete statistics, but I cannot find an unbiased estimator that is function of $T_n$ nor the MVE estimator. (I actually have trouble even in finding any unbiased estimator of $\theta$, so that I cannot use Rao Blackwell theorem). Thank you for your help.