On my configuration (VM, Ubuntu 18.04 server, tomcat9) I have found tomcat stdout messages in the /var/log/syslog file:
root@deploytest:/var/log# less syslog
Dec 4 07:17:43 deploytest systemd[1]: Starting Apache Tomcat 9 Web Application Server...
Dec 4 07:17:43 deploytest systemd[1]: Started Apache Tomcat 9 Web Application Server.
Dec 4 07:17:44 deploytest tomcat9[3235]: Server version name: Apache Tomcat/9.0.16 (Ubuntu)
Dec 4 07:17:44 deploytest tomcat9[3235]: Server built: Sep 11 2019 19:47:51 UTC
Dec 4 07:17:44 deploytest tomcat9[3235]: Server version number: 9.0.16.0
Dec 4 07:17:44 deploytest tomcat9[3235]: OS Name: Linux
Dec 4 07:17:44 deploytest tomcat9[3235]: OS Version: 4.15.0-72-generic
Dec 4 07:17:44 deploytest tomcat9[3235]: Architecture: amd64
Dec 4 07:17:44 deploytest tomcat9[3235]: Java Home: /usr/lib/jvm/java-8-openjdk-amd64/jre
Dec 4 07:17:44 deploytest tomcat9[3235]: JVM Version: 1.8.0_222-8u222-b10-1ubuntu1~18.04.1-b10
Dec 4 07:17:44 deploytest tomcat9[3235]: JVM Vendor: Private Build
Dec 4 07:17:44 deploytest tomcat9[3235]: CATALINA_BASE: /var/lib/tomcat9
Dec 4 07:17:44 deploytest tomcat9[3235]: CATALINA_HOME: /usr/share/tomcat9
...
Dec 4 07:18:01 deploytest tomcat9[3235]: Starting ProtocolHandler ["http-nio-8080"]
Dec 4 07:18:01 deploytest tomcat9[3235]: Server startup in [16,667] milliseconds