It's straight-forward to log slow queries in MySQL both on your local machine and on an Amazon AWS RDS instance. I'll walk through both, although MySQL has good documentation on slow query log and log destinations.
There are many blog posts about enabling slow query logging to a file within the MySQL data folder, but very few point out that, for files outside that folder, you have to manually create the log file and set its permissions correctly with touch and chown.
AWS does not allow shell access, so there is no way to read a local log file. Instead, RDS logs the output (log_output = TABLE) to the built-in table: mysql.slow_log. Here are the RDS command line instructions for setting up the parameter group, modified from Inventables' excellent write-up.