Infinite multiplication

There is no simple criterion that is necessary and sufficient, but I can give some examples. If the n-th term in your sequence is e^{-1/n}, then the log is -1/n, so the sum of the logs diverges (it is the negative harmonic series) and M=0. If the n-th term in your sequence is e^{-1/n^2}, then the log is -1/n^2, and so the sum of the logs converges and M is not 0. IIRC, the sum of -1/n^2 is -pi^2/6 [aka -zeta(2)], and so M=e^[-pi^2/6] for that example.

There are a variety of tests for whether or not a series converges--see your favorite calculus textbook. One trick that might be useful in dealing with examples is that log(1-x) ~ -x for small x.