I have a function of $X_1,...X_n$ iid random variables with pdf
$$f(x|\theta) = \frac{\log\theta}{\theta - 1}\theta^x, 0<x<1$$ and I am trying to find the maximum likelihood estimator of $\theta$. I am also given four $x$ values to plug in at the end in order to determine a numerical estimate.
I tried solving this by taking the log of the function, taking the derivative with respect to $\theta$, setting the derivative equal to $0$, then solving for $\theta$. However, when I did this, I found that solving for $\theta$ became extremely complicated. Is there an easier way to go about solving this without taking the log?
I came up with the log of the likelihood function as being: $$ \sum \left[ x_i\log(\theta) - \log (\theta - 1) + \log(\log(\theta)) \right]. $$
I am stuck on where to go from here because taking the derivative and setting equal to zero becomes something that is very difficult to solve for $\theta$