So I just fixed an interesting bug in the following code, but I'm not sure the approach I took it the best:

p = 1
probabilities = [ ... ] # a (possibly) long list of numbers between 0 and 1
for wp in probabilities:

  if (wp > 0):
    p *= wp

# Take the natural log, this crashes when 'probabilites' is long enough that p ends up
# being zero
try:
    result = math.log(p)

Because the result doesn't need to be exact, I solved this by simply keeping the smallest non-zero value, and using that if p ever becomes 0.

p = 1
probabilities = [ ... ] # a long list of numbers between 0 and 1
for wp in probabilities:

  if (wp > 0):
    old_p = p
    p *= wp
    if p == 0:
      # we've gotten so small, its just 0, so go back to the smallest
      # non-zero we had
      p = old_p
      break

# Take the natural log, this crashes when 'probabilites' is long enough that p ends up
# being zero
try:
    result = math.log(p)

This works, but it seems a bit kludgy to me. I don't do a ton of this kind of numerical programming, and I'm not sure if this is the kind of fix people use, or if there is something better I can go for.