Is there a Play Store log file?
Or should I be looking where apps are installed on the phone for time/date stamps?
TitaniumBackup also displays this when you go to the batch (backup/restore) tab. You can have it sort your apps by date of installation, name, size and so on.
It should be noted that you need to root your phone to do so.
It would give you a humungous list of details about all the installed packages including the date/time stamp you're seeking. If you want the said stamps for a specific package whose package name you're aware of, then enter:
Its output possibly would be like,
firstInstallTime=2015-07-04 15:49:50 lastUpdateTime=2015-07-04 15:49:50
Note that the info is originally available in the file
Alternatively, if you've Google Play Store installed, then take its backup using ADB, extract the backup to locate the database
This approach has its limitation. Play Store wouldn't show date/time stamp for sideloaded apps.
It is possible that the former approach also gives unexpected result for some apps. In that case, do further research!
I didn't find a place in the stock-system which tells this. But there are useful tools around like AppMonster which display this information along (besides, AppMonster allows you to backup your apps, and in the Pro-Version even does so automatically on install as well as update -- so you can revert to a previously installed version easily in case of trouble with an update).