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> the log of the density can't have an upper bound at 0

I don't think he claims that. He even shows charts where positive log-densities appear.

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.

The log function has a vertical asymptote as x approaches 0.

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.

If the author himself tells you so, it’s more than likely. :-)

When x goes to minus infinity log(x) approaches zero.

No, the logarithm is only defined for positive values. Plus probability densities are strictly non-negative. https://en.m.wikipedia.org/wiki/Logarithm

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.

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