Our log files are under <serverinstall>/usr/logs, as you suggest. Actually, they are under <serverinstall>/usr/logs/<hostname>, I assume because we have a cluster of two ArcGIS Servers set up. It's the same on both servers; although the hostname is different, of course.
If you do not have a logs directory under usr, that suggests they are configurable in some way; at least in terms of their location. Hopefully, in terms of their permissions also.
Cheers,
jtm