Log Magento M2E Pro execution to check it

I'm using M2E Pro on a test environment (a cheap hosting service) to try it and develop our own platform to sync a website with Ebay and Amazon.

So far syncing seems to work good, but since I've understood that M2E Pro syncing performances depend on the host too, I'd like to track the sync cron job to check

1. When it's executed


2. When it ends


3. If any error occurs

How can I do that?