How is rationale used in a sentence? Can it be used in place of logic?
I would think that rationale could rather be replaced by reasoning than logic, but both are certainly possible. Rationale highlights more the deduction process than the set of rules behind it.
Rationale is derived from Latin ratio, which is the human reason, brain, wit, understanding.
Look at rational, rationality: in philosophy, rationality is the exercise of reason, a key method used to analyze the data gained through systematically conducted observations (wiki). In economics and such, a rational decision is one that is optimal given the available information.
Therefore, a sentence using rationale would list the information and reasoning that led to a decision.
Rationale can sometimes be used in place of reason or logic. (It is not a drop-in replacement that would work in all cases)
Examples from the American Corpus: