I don't think that's normal for eleukes at all. When nothing is plugged in, regardless of what your volume/gain settings are, the uke is 'off'. When something is plugged in - either an amp cable and/or headphones - regardless of the knob settings, the eleuke is 'on' and power is draining whether you're playing it or not. Always unplug after use.
The only time I drained my battery was when I accidentally left a cable plugged into the uke after playing for maybe 1/2 hour, and didn't realize it til about one-two days later. When I realized it, I could play maybe a few chords amped before the battery completely died out. In that estimation, the battery should last for at least 24-48 hours of continuous plugged in time. If you play only 1/2 hour/day, and the battery is gone in 3 days, that's only 1.5 hours of battery use. Even if headphones drain more battery, 16 times more battery power than using an amp sounds a bit far fetched.
ETA: though, to be fair, I have a older model of Eleuke (before they have the additional mp3 player plug in in addition to the headphone plug in), so maybe it's the newer models?