There is a description of what MySQL does here. It takes the key almost as is and applies AES in ECB mode, which is weak since it leaks which blocks are identical to each other in data encrypted with a given key. You should not assume that data which was encrypted with MySQL's AES_ENCRYPT() is really "secure".
That being said, whether your key and data will make it to the binary log depends on how it was configured. If using statement-based formatting, then the binary log will contain the SQL statements which were issued (at least those which entail modifications), and your data and key may be revealed that way. If you use row-based formatting, then the log contains the results (alterations to actual rows), and your cleartext data and key won't appear in the log. See this question for pointers.