Is there a way to access SFDC API log?

There is now the Event Monitoring API (Paid Feature). You can use this with the API EventType to get a complete log of all the calls that were made on a certain day. You can use the METHOD_NAME, USER_ID, CLIENT_ID to identify who and what was making each call.

I've put some additional details about this in Monitoring your Salesforce API usage

In case you need more general overview, it can be useful to see API Usage Report. You can find it under Reports->Administrative Reports->API Calls Made Within Last 7 Days. It helped me quickly locate a source of an excessive amount of API requests.

There is no specific log per API call. However, you can enable the debug log for the user that the application is using to log in as under Setup->Monitoring->Debug Log. That will track up to 20 API calls (or inserts, updates, deletions, VisualForce pages, etc..). If you're actively monitoring it you can continually click the reset link to reset the limit and capture more logs.

If that's more detail than you're looking for and the application logs in each time it needs to use the Web Service API (rather than maintaining a session) you can simply look at that user's login history at the bottom of the user page. Look for a login type that isn't "Application" or "Sync".