Logic doesn't provide morals.
Simple eample is a small child - he's usually way too selfish and must be taught to share and wait while others can have this thing.
But any child has logic and this very logic makes him selfish - "why should I share? It's MINE!"
If logic would prevent people from crimes - there would be no crimes.
Do you say that 99% of people are stupid and without logic???
Usually the worse criminals are also of the cleverest people - but they have no moral that would prevent them from misusing their gift.
It's simple - if you have nobody to fear (and you can always bribe a cop or a judge since they are also immoral) - then you have nothing to stop you from crimes.
The key point that makes religion a good source for moral is the concept of "everything you do is seen on High".
I'm not speaking about the "people of religion" who are still people and might err (even a lot).
I'm speaking about the religion itself - the idea of being observed by God constantly.
This gives you the REASON to act morally.
An example which isn't very on-topic but can shed some light on the idea:
Once a Rabbi was put in jale by anti-semits.
One of the interrogators threatened him with a gun.
"Be quiet or I'll use this "toy" on you!"
The Rabbi answered:
"This "toy" can threaten a man who has many gods and one world but not a Jew like me who has One God and two worlds."
So you can see that whenever a person "thinks for himself (selfishly)" he can "logically" assume that even killing is "moral" if HE is not the victim.
Logic isn't the source of moral - it's the source of destruction of moral.
