I would say the exponent of a variable makes (contributes to making) the result large.

Given that a negative exponent makes a result smaller, I'd say that it describes rate at which it'll change.

What if the base is less than one?

I think the question "does it make it larger or does it describe how it gets large?" implies that the transformation is increasing.

My intention was to say that the exponent describes the characteristics of a change, not to imply that the effects are only positive.