Are there logs written to when issues are encountered with the API?

To ensure issues are adequately captured when errors arise, it's recommended to implement a logging framework to output any errors encountered into a file that can be reviewed. Exceptions that are captured in this way are more effective in troubleshooting issues that could occur.
Many Logging frameworks can be considered, such as ILogger for example. More information regarding logging can be found here: Logging in C# - .NET | Microsoft Learn.

KBA is applicable for both Cloud and On-premise Organizations.