Normally when you run ntbackup logged on as “Administrator” backup logs are saved in “C:\Documents and Settings\Administrator\Local Settings\Application Data\Microsoft\Windows NT\NTBackup\data“
If backup logs are not in the expected location check following:

If you run ntbackup from the command line or a script make sure you include log option: /l:s (summary log) or /l:f (full log).

If you run ntbackup as a scheduled task, check “Run As” option in scheduled task properties. For example if you run the scheduled task as “backup-operator” your ntbackup logs will be saved in “C:\Documents and Settings\backup-operator\Local Settings\Application Data\Microsoft\Windows NT\NTBackup\data“.

Windows Server 2003