It's not that the imaginary parts are small - they're defined on a range 2π wide (if you're working in natural logs, and some weird width otherwise). It's that they're hard to interpret, and if you know (and need to know) enough to be working with them there's little point to taking the log.
It's well-established that 0 decibans means 1:1 odds or 50% confidence; that 10 decibans means 10:1 odds; that -10 decibans means 1:10 odds; and that fractional numbers of decibans have similar meaning.
Does it make sense to talk about "i decibans", or "10 + 20i decibans"? If so, what does that actually mean?
I'm currently roughing out what may eventually become a formal specification for a protocol. It includes a numerical field for a level of confidence, measured in decibans. I'd like to know if I should simply define the spec as only allowing real numbers, or if there could be some purpose in allowing for complex numbers, as well.