Unfortunately No, it is not logged.
The task is set to started in the database and the scheduler service picks up that there is a newly started task and handles it.
It is all happening by the a mix of Console code and it sending SQL statements to the database.
So at the time the task is started, we know who is doing it at the moment it happens, and we make sure they have rights to do the action, but there is no storage of this information. It passed in queries through the database to verify the user can access the task and start it and then allows them to start it and then the information is not stored and that information is gone as soon as it happens.
So you only chance would be to somehow capture the SQL queries at the moment they happen and store them, and even then I am not sure if you could determine the user who initiated the task.