Exponentiating makes everything multiplicative that was additive before. This rule of course applies to monotonic effects as well. This has to be kept in mind.
If b is the range of the monotonic effect on the log-scale, then
\exp(\eta + b) - \exp(\eta)
is the range on the expoentiated scale where \eta is the rest of the predictor term (i.e. other predictors) you condition on.
When transforming summary statistics, please keep in mind that this does not work for all statistics. For instance, the mean is not equivariant under non-linear transformations.
It is best to transform the posterior samples and then summarize only after having applied all transformations.