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