In this example of a rc.local script I use io redirection at the very first line of execution to my own log file:
#!/bin/sh -e
#
# rc.local
#
# This script is executed at the end of each multiuser runlevel.
# Make sure that the script will "exit 0" on success or any other
# value on error.
#
# In order to enable or disable this script just change the execution
# bits.
#
# By default this script does nothing.
exec 1>/tmp/rc.local.log 2>&1 # send stdout and stderr from rc.local to a log file
set -x # tell sh to display commands before execution
/opt/stuff/somefancy.error.script.sh
exit 0