This is no standalone app. You need to install IP Widget (play.google.com/store/apps/details?id=de.dieterthiess.ipwidget)
Log your external IP address. Always know which ip you had. The data is only stored in a local database, so your data will not leave your device.
If you want to monitor your external IP address changes while your device is connected to the same wifi all the time, you need to enable the manual update in IP Widget settings, so the device will update data periodically and will detect changes of the external ip address.