How can we access salesforce API logs?

You can use EventMonitiring API, a paid feature, to access a complete log of all the calls made for a day. To get a more general overview, you can see the API Usage Report from Reports->Administrative Reports->API Calls Made Within Last 7 Days. Another way to check this is to enable the debug log for the user the application uses to log in as under setup ->Monitoring->Debug Log. It can track a maximum of 20 API calls. Click the reset link to reset the limit and capture more logs.