If Not, then no, unfortunately. However, you could always sign up for one of these services. Let's talk about how to monitor your downloads once you have.
Once you're paying for Google Apps Unlimited or signed up with Google Apps for Education, you have access to something called the Google Apps Admin Console.
From the Google Apps Admin Console, click Reports, then Audit, then Drive. From here, you'll be able to see all kinds of information, including what action was performed, by what user, at what time and date, etc. Note that the Drive Audits only last for up to 6 months in retention time.
On the side of the page, there's a Filters section that you can use to create your own filters. From here, you can create a filter for users downloading files, and you'll be given the user's name, their IP address, the exact date, and more.
The possibilities offered by the Drive audit log doesn't stop with seeing who downloaded what, either. Other information you can find is who has created, viewed,, previewed, updated, downloaded, shared or even deleted content on Drive. This doesn't stop with Docs- it covers other Google Apps, and even files that you upgraded to your Google Drive.
Since you're using Google Apps Unlimited/For Education, you now have access to a whole bunch of new applications and features. In addition to the Google apps and services you're already used to, like Hangouts and Drive, you also have access to things like Google Sites, the Admin Console, and Google Vault.
In particular, the Admin Console adds a lot of extra functionality to the sharing functions in Google Apps, in addition to insights and reporting and free support. Meanwhile, the Vault is an add-on archive service that saves all business communications, should you need them.
To learn more about what you can do with Google Apps Unlimited and Google Apps for Education, read our article on the matter. For more on the latest in education and tech, browse our blog, and contact us for a consultation if our services could help you.