On Tomcat 9, you can view logs by the command
journalctl -u tomcat9.service
This will display logs in a viewer like "less".
To follow logs, use the command
journalctl -f -u tomcat9.service
To write the full log into a file, use the command
journalctl -u tomcat9.service > tomcat-log.txt
If you're using an older version (Tomcat 8), you can get the log file by the following path:
/var/lib/tomcat8/logs/catalina.out