Suppose $X_1,…,X_n$ are iid with density $f(x|α)=(α+1)x^αI(x∈[0,1])$, where $α>−1$ parameterizes the family.
Suppose $x_i∈[0,1]$ for all $i$. I am trying to find the natural log of the density (log likelihood) given the parameter $α$ as a function of $x_1,…,x_n?$
The density for a sample is
$$f(x|a)=\prod (a+1)x^aI(x_{min}\ge0)I(x_{max}\le1)\\ =(a+1)^n\prod x^aI(x_{min}\ge0)I(x_{max}\le1)$$
Take the log to get the log likelihood
$$\log f(x|a)=n\log(a+1)+a\sum\log x+\log I(x_{min}\ge0)I(x_{max}\le1)$$
(it is undefined if there is a data point not in the support)