I don't think there is an exponent curve as such for battery tech. Li-ion came in about 2006? And nothing much has improved since then. The trouble with batteries is you can't just shrink components and get some improvement as you do with semi-conductors. Your components are already on the atomic scale. So more fundamental breakthroughs are needed.

The prediction is based mainly on our increasing control of biology and the ability to work on the small scale. If nothing else we'll invent a way to metabolise fat or other carbohydrates to electricity and have small home bioreactors that produce carbs and make nice little cartridges for people to plug into their electronics. Maybe not in 10 years, but some substantial movement is definitely possible in this direction.