The closest you get to this logging is Audit Trail. However, this does not hold any information about your API requests towards endpoints in SFMC.
When it comes to API/integration monitoring, is also not dealing with this on the receiving end, but more on the sending end. Your external system should be accountable for maintaining the integration, error handling, retries, etc. Errored requests which are a result of failed authentication, or incorrect endpoint URL might not even reach SFMC.
Your best way of monitoring whether you are successful in receiving data is to have each request populating a timestamp field in the data extension. You could then query this data extension on e.g. hourly basis, to validate if you have received any updates. Querying records updated since last check, into an additional data extension, would allow you to use Verification Activity to alert someone, if no records have been updated (record count in this addional DE is 0)