cron will call syslog to report stats of cron jobs. Where syslog puts the messages depends on your exact setup. For me, cron status log messages go in /var/log/syslog, but you might have a cron specific logfile in your /var/log directory.
Often, cron will send a local email to the root user with cron errors. You'll only see this if you have local mail delivery configured so you get mails for root.
The quick-and-dirty method for seeing error messages would be to change the crontab line to:
05 * * * * /db/app/test.sh > /tmp/mycronlog 2>&1
...and wait for the program to run again, then look in /tmp/mycronlog.
General hint about cron problems: it's almost always because of the environment settings in general, the PATH variable in particular. When cron launches a job, it does so with a very restricted environment, as defined in the /etc/crontab file.