Create a view of the 'Recent Log Messages'
In Drupal 8, you can simply create a view of "Log entries". Just make sure you add a "Type" filter and select "user" as the type.

Then under "Page settings", change the "Access: Permission" to "Role > Administrator". However, if you don't want this user to view other log messages, perhaps that user shouldn't be an administrator? I recommend creating a new role for that type of user.

In Drupal 7, the Views module doesn't allow you to make a view of log entries out-of-the-box, however if you install the Views Watchdog module, you can do it just like the Drupal 8 method described above.