In which directory are system log files kept?

/log/syslog/ does not exist as a defined path in the Linux FHS and is not created by any standard distribution.

/var/log/ is defined by the FHS as the standard location for log files and directories, containing files such as syslog, auth.log, kern.log, and application-specific logs. All major Linux distributions comply with this standard, making /var/log/ the correct and universally expected location.

/sys is a virtual pseudo-filesystem that exposes kernel objects and hardware information at runtime; it contains no persistent log files.

/var/log/sys/ is not a standard sub-directory; log files reside directly under /var/log/ or in application-specific sub-directories like /var/log/apache2/, not under a generic sys/ folder.