There is no common api, this is why asp.net core uses an injected logger.
The Unix convention is to write to a log file (stdout or path passed as arg) . Typically to folder in /var/log/
You can also log to system logging (typically /var/log/system.log) by using posix api openlog() and syslog() (use pinvoke from c#)
As you are probably going to run the asp.net core app with systemctl or systemd depending on Unix/Linux variant, you will want a logging provider that can write to a file.
Your distribution may also have a log journaling feature. It will have its own api.