I have this term $ \prod_{i=0}^N(u_i-x_i-\tau)^{-3/2} $ and on which I need to take the logarithm.
Applying the log make the $\prod$ to $\sum$.
What bothers me is the power $^{-3/2} $.
Will it be
(a) $-3/2\sum_{i=0}^N (u_i-x_i-\tau)$ or will it be
(b) $\sum_{i=0}^N (u_i-x_i-\tau)^{-3/2}$
Neither of them: if $u_i-x_i-\tau > 0$ for each $i=0,\dots,N$, then
$$\log \prod_{i=0}^N(u_i-x_i-\tau)^{-3/2} = \sum_{i=0}^N \log (u_i-x_i-\tau)^{-3/2} = -\frac{3}{2}\sum_{i=0}^N \log (u_i-x_i-\tau).$$