It's possible that I misunderstood the text but what's the alternative interpretation of "The log function asymptotes at 0"? Also note that the text says log probabilities when it actually refers to log densities.
Ah, that's likely the interpretation that the author had in mind. But it didn't help that he wrote log probability when he actually talked about log probability density.
You're absolutely right! I meant it the other way around:
As x approaches 0, log(x) goes to minus infinity.
I think that's what he means (the next sentence about "visualiz[ing] very small values of a function by expanding their range" makes sense in that context), but I agree it is not very clear.
I don't think he claims that. He even shows charts where positive log-densities appear.