Logarithmic Randomness is Necessary for PCP Theorem
You can actually show a slightly stronger result: if $SAT \in PCP(o(\log n), o(\log n))$ then P=NP. The idea is to apply the implied reduction repeatedly, reducing the size of the instance all the way down to $O(\log n)$, at which point you can solve it by trying all certificates.

For more details, Arora and Safra (page 76) suggest looking at Theorem 26 in the appendix of FGLSS (journal version). Unfortunately, this appendix doesn't seem to be available any more.