You can get a daily log file from iTunes after a sync to list each app activation that happened in the past day.
If iTunes offers to erase content - say no and just set it up to do a basic sync (not syncing any content in the many tabs). Once the sync is complete, you will have copies of the system logs. Should you not be able to do this, there are also programs like PhoneView that will peep into the device without needing a sync.
On a mac, look in ~/Library/Logs/CrashReporter/MobileDevice for the name of the folder matching the rental device. Look for files named log-aggregated-2011-11-14-000000_loanerPad.log and you can see a daily summary of the amount of time each app is active as well as how many times it was launched.
Presumably, you will open each app you wish to record to get all of them listed in this one file for later reference. You will only find the internal name of the app, so a bit of googling (or getting screen shots) might be needed to find the app store name of an app if it's internal name isn't the same as what you expect in iTunes. I would also use the screen capture (and expand each folder as needed) just to be sure you have all the information you need to find these apps at a later date.
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