By default the Sauce Connect output will tell you where the log is. Look for the this line:
Log file: <some/path/to/a/file.log>
If you do not have access to the STDOUT you can define the log name and location when the tunnel starts! The flag is --logfile or the short-flag -l. Read more here: https://wiki.saucelabs.com/display/DOCS/Sauce+Connect+Proxy+Command-Line+Quick+Reference+Guide
If you only provide a filename, Sauce Connect will create that file in the same directory as the Sauce Connect executable. So check that directory for the newly created log file.
$ ./sc-4.5.4-osx/bin/sc -u sauce_user -k **** --verbose --logfile qa-local-test.log