For On-Premise Exchange you are looking forMailbox Transport service logs which sit in
%ExchangeInstallPath%TransportRoles\Logs\Mailbox\Connectivity
but you have to explicitly enable them first.
This article will get you started
http://technet.microsoft.com/en-us/library/bb124500(v=exchg.150).aspx
You can't really do that in O365 without raising a ticket with MS, the only available cmdlets are: