I"m having some trouble with a question from my statistics class. We have to find the MLE of $\theta$ where $X_1,...,X_n$ are iid random variables from the density
$f(x|\theta)=\theta(\theta+1)x^{\theta-1}(1-x)$
with $0\le x\le 1$ and $\theta>0$.
I've tried taking the log-likelihood, and after deriving I get
$\frac{\partial}{\partial\theta} L(\theta|X_1,...,X_n)=\frac{n}{\theta}+\frac{n}{\theta+1}+\sum_{i=1}^n \log(x_i)=0$
which is a bit hard to isolate $\theta$. But also, isn't $\log(x_i)$ potentially undefined for $x_i=0$?
Thanks a lot!