There is no official tutorial on how to do that, but I will add it soon for sure.
In general, you will need to either fire an event and then create a listener or subscriber which will listen for that event and then log the message. Check "app/Listeners/UserEventsSubscriber.php" file to see it in action.
Now, if you don't want to follow that approach, you can simply use the code below anywhere in your controller to log a specific action:
app(\Vanguard\Services\Logging\UserActivity\Logger::class)->log('Your custom log message here');
This will automatically write the log message you provide into the activity log for currently authenticated user.