= Solution
For the <independent and identically distributed> observations, multiply the <probability density functions>. On the admissible parameter range $\theta>-1$, the <likelihood function> and <log-likelihood> are
$$
\boxed{\mathcal L(\theta)=(\theta+1)^N\prod_{i=1}^Nx_i^\theta},\qquad
\boxed{\ell(\theta)=N\log(\theta+1)+\theta\sum_{i=1}^N\log x_i}.
$$
These expressions presume that every observation lies strictly between zero and one; an observation outside that support would give zero <likelihood> under this model.
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