System Wide LoggingI found a solution in Ubuntu which was to open up /etc/default/bootlogd and set the option to yes. You supposedly see the boot logs in /var/log/boot. However, I found it was writing to /var/log/boot.log. Sort of. I got partial output (about 6 lines). The file was never fully written to.
Script Specific LoggingI found a workaround. Instead of using a system wide log, use a script specific log. Add this to your script. It will log all output.
exec > /tmp/debug-my-script.txt 2>&1