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ServiceAPI is just another Episerver site so normal EPiServerLog.config should work. However from the symptons it looks like a performance degradation regarding memory usage. I would suggest to use some profiling tool to monitor memory consumption. You can always contact our developer support service for further assistance.
I can name a few, jetBrains dotMemory, ANTS .NET Memory Profiler. WinDbg is also a powerful tool - but it can be complicated to use.
Service API
We have a schedule job, that take a lot of items (loop) and add into Commerce with service API, but after a while, the job start to have problem to get response from ServiceAPI, each call is taking more and more time to get a response, until I get Aplication_Start in log, and job hangs.
Is there any way to activate logging in Service API?