Is there a Play Store log file?
Or should I be looking where apps are installed on the phone for time/date stamps?
Many apps available from Play Store show the install and update time of the installed apps. I've tested three of them and have given instructions to use them appropriately.
After installing and launching it, go into Applications → tap on your app's entry → View Details.
Install and launch the app → tap your app's entry → General.
Install and launch the app → from the left sidebar select Applications → User apps → tap on your app's entry.
(Click an image to enlarge it; order of images: Elixir 2, App Detective and Solid Explorer)
If you want a command-line way, follow these instructions:
Setup adb in PC and execute the command:
adb shell dumpsys package
You might end up with overwhelming information 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:
adb shell 'dumpsys package PKG_NAME | grep -e "firstInstallTime" -e "lastUpdateTime"' # replace PKG_NAME with the package name of your app
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 /data/system/packages.xml.
Alternatively, if you've Google Play Store installed, then take its backup using ADB, extract the backup to locate the database localappstate.db. Open that file in PC using an SQLite database viewer like DB Browser for SQLite or on Android using apps like SQLite Editor / aSQLiteManager. The said apps may need root access so I recommend to stick with the SQLite viewer for PC.
The table appstate in that file has the columns first_download_ms delivery_data_timestamp_ms. The latter shows when the app was installed and the former shows when the download was initiated by Play Store.
Limitation: Play Store wouldn't show date/time stamp for side loaded apps.
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).