You could try to schedule a cron job in the crontab file to call a script every 5-10 minutes that checks for any changes in your logs and sends you an email if an error entry is found.
From the crontab documentation:
The crontab command invokes an editing session that allows you to create a crontab file. You create entries for each cron job in this file. Each entry must be in a form acceptable to the cron daemon. For information on creating entries, see The crontab File Entry Format.
Exanple:
To run the calendar command at 6:30 a.m. every Monday, Wednesday, and Friday, enter:
30 6 * * 1,3,5 /usr/bin/calendar
Alternatively, you can create a crontab file by specifying the File parameter. If the file exists, it must be in the format the cron daemon expects.