= Solution
Put $S=\sum_{i=1}^N x_i$. Since the observations are <independent random variables>, their joint <probability density function> is a product. Therefore the <likelihood function> and <log-likelihood> are
$$
\boxed{\mathcal L(\theta)=\theta^{2N}e^{-\theta S}\prod_{i=1}^N x_i},\qquad
\boxed{\ell(\theta)=2N\log\theta-\theta S+\sum_{i=1}^N\log x_i}.
$$
These expressions assume positive observed values, as occurs with <probability> one under this continuous <Erlang distribution>.
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