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. Please note that the syslog option is only supported on Linux and OSX environments.
Additionally, you can specify the location of where you want the logs saved, the number of bytes that should be written to a log before it needs to be rotated, the duration a log should be written to before it needs to be rotated, and the maximum number of older log file archives to keep using the -log-file, -log-rotate-bytes, -log-rotate-duration and -log-rotate-max-files command line flags respectively.
As an alternative to logging, you can connect to any running Consul instance with consul monitor to view operational logging at any level from err to trace so that is something to keep in mind too.