Definitely 'as'. 'Like" is incorrect.
EDIT: I will try to explain the reason. I am not a grammarian, if anyone who's actually made a study of this sort of thing would chime in, I'd be interested myself.
'Like' is used for comparisons. You could use it in the following: "This happened at the same time before. Today is like yesterday."
'As', as an adverb (according to the dictionary), is used to relate the 'extent' of something, which is a little difficult to fit exactly into this situation. But in your sentence you are not comparing the two things; you are saying that, under some conditions, the extent of one thing is the same as the extent of the other.
As a native US English speaker, I find the explanation a little hard to hold onto. It is easier, for me, to understand the part about comparing two things than to state why this situation does not qualify as comparing two things. But that's as far as I've gotten with it, hope it helps.