Inverter question

Hello,

Further to my previous questions around batteries and solar panels.

I have now 2 x 100 ah deep cycle gel batteries.
150w fixed panel, 180w portable panels through the 30 amp controller.

I am in the process of buying a 2000w pure sign wave inverter, (that size to run the English Rose,s hair drier, [don,t go there]).

So the question, finally, is, does a large output inverter running low powered equipment use more power to run then a smaller inverter running the same low power load.

I hope that makes sense.

William