Consul logs to standard output which can be redirected in your
startup/init/systemd unit file or to any file you choose. That said, there is also a -syslog command line flag and a corresponding enable_syslog configuration file option that will enable Consul to log to syslog as well.
As an alternative to logging, you can connect to any running Consul instance with consul monitor to view operational logging at any level from
trace so that is something to keep in mind too.