Why are the Recent log messages not shown within admin/reports?
The "Recent log messages" is one of the submenu options located within admin/reports/dblog.

It becomes available after the core module dblog (Database Logging) is enabled.

If that core module is enabled, and it still isn't showing up, then it is probably because permission "View site reports" (within "System") has not been granted to the user trying to access that report.