Does cron produce an output report of what has been run?

In an earlier post of mine I mentioned that drupal was very slow if the site hadn't been visited for a while etc.

It seems a big reason for this is cron running if the site hasn't been visited in a while.

Anyway, after lots of digging around I have come across a number of recommendations which suggest setting the cron function to never on the config/system/cron page

Then setup an external cron function on cpanel etc to have the server run it instead of the site itself.

I am trying to ensure I receive notifications that the cron function runs at the specified intervals each day by email.

However, my hosting company said a reason I wasn't receiving an email about this it time cron runs was because the cron job did not produce and output and that is why the cron job I set up is not sending any mail.

So does running cron produce a report of any kind about what it does?

How can I see this report if it exists?

How do I ensure I receive a notification that cron is running at the intervals it has been set to run?

Basically all I want to do is ensure cron is running and that by doing it via the server instead of via the drupal cron page, it won't affect a visitor by being a really slow page.

Thanks,
Sam