I'm using the Android Kindle App, and notice that it stores its content in the old standard for the SD card, /mnt/sdcard. On modern Android devices, though, that location is typically actually part of internal memory (which allows manufacturers to avoid having to ship devices with an SD card installed, probably saving some money on each device). It's therefore become typical on Android apps which may use a lot of local storage to allow the user to switch storage to the real SD card. Since the directory path is usually unique to each device, it typically requires the user to enter the path to the real SD card, and then move the data there. Overdrive does this. Podcast Addict does this. Any chance the Kindle App can be changed too do this?
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