My original response to this was wrong and has been deleted

I don't think this has anything to do with logs, but rather that it is about the difference between probabilities and odds. Specifically, the Bayes factor works on the odds scale but the proof for conservation of expected evidence is on the regular probability scale

If you consider the posterior under all possible outcomes of the experiment, the ratio of the posterior probability to the prior probability will on average be 1 (when weighted by the probability of the outcome under your prior). However, the ratio of the posterior probability to the prior probability is not the same thing as the Bayes factor.

If you multiply the Bayes factor by the prior odds, and then transform the resulting quantity (ie the posterior) from the odds scale to a probability, and then divide by the prior probability, the resulting quantity will on average be 1

However, this is too complicated and doesn't seem like a property that gives any additional insight on the Bayes factor..