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In the troubleshooting section of the admin interface (SM16), there is an option to change the logging of the API ("API Service") however I do not see any log file in either of the two folder below that relate to API calls:
C:\Program Files (x86)\SmarterTools\SmarterMail\MRS\App_Data\Logs
The failed authentication attempts against the new API seem to be feeding into the Administrative log (as the webmail, logically given it is the same interface), which is good. But I see nothing in any log about failed authentication attempts in the old API (SOAP Service).