Does SQL Server has an external log file or internal table for attempted connections, or is that kind of info put somewhere in the Windows Event Log?
Another way to check on connection attempts is to look at the server's event log. On my Windows 2008 R2 Enterprise machine I opened the server manager (right-click on Computer and select Manage. Then choose Diagnostics -> Event Viewer -> Windows Logs -> Applcation. You can filter the log to isolate the MSSQLSERVER events. I found a number that looked like this
Login failed for user 'bogus'. The user is not associated with a trusted SQL Server connection. [CLIENT: 10.12.3.126]