Logic by itself isn't sufficient for this argument. As you say, someone could offer 50,000 logical arguments against the existence of God, and it wouldn't prove anything. The converse is also true, someone else could offer 50,000 logical arguments supporting the existence of God, and it wouldn't prove anything either.
If you want to show that God exists, you must be able to produce evidence to that end. This burden rests on you and others like you, because it's impossible to prove that something doesn't exist.