First see here how to make an "Action Config to run one instance of a task". This couples a log event code to an action config. When a log message with the configured event code is logged and reaches the central database, the action config is triggered. The tag in the action config has to refer to a MC:_SystemAlerter task that can send emails.
Then use the info here to create a new MC:_SystemAlerter task. For the new task use the tag from the action config and make it a one-off task. The xml file that needs to be uploaded contains the settings for the emails (see mcSystemAlerter.xsd schema). In here it is possible to use the tag %LOG% in the body of the email. This tag will then be replaced by the logmessage(s) that triggered sending the email.
...