If you don't have root rights then the log file, if it resides in /var and has sane access permissions, will be inaccessible to you anyway. As far as an example goes something like
Code:
while read LINE; do [ "${LINE//error/}" != "${LINE}" ] && { echo 'Got one!'; }; done < <(tail -f /var/log/messages)
could do looking for the term "error" in /var/log/messages.