Patrick McGee, Wolfram Research supplied the following nice use of Reduce to explain what I should have thought of ...
In[1]:= Reduce[Log[a b] == Log[a] + Log[b], {a, b}, Reals]
Out[1]= a > 0 && b > 0
Thus, if a or b is zero, or negative, it is no longer true that the logarithm of a product is the sum of the logarithms, obviously.