Which distro i.e. operating system is there in webserver? To check, login on webserver and invoke:
As far as I remember, a normal user can read common system logs. But as I said which distribution is there.. perhaps then people can better help you.
Also do you want to check script run logs for yourself or any other user?
FYI, system logs contains many type of log files (in Linux, common logs are maintained in /var/log/.... directory. In Solaris it's in /var/adm/... directory). Also a webserver may maintain it's logs somewhere else (I am not sure).
In the meantime, you can try:
grep <username> /var/log/*
grep <username> /var/adm/*
One more thing, /home/username
/.bashrc_history, contains a list of all commands user has invoked, so you can check that also.