Inverted U-shape estimation


I am currently writing my master dissertation about the impact of corporate income tax reduction on the competitiveness of nations. It might be probable to assume that a reduction of corporate income tax will have a certain effect depending on the level of corporate income tax already imposed in a country on a particular moment. An inversed U-shape might be considered where a reduction of corporate income tax might first result in an increased competitiveness of institutions since it lowers the regulatory burden. After a certain point, it might result in a decrease of competitiveness when corporate income tax is lowered to a level where the governmental income is too low to impose a decent regulatory framework.
The investigation will be elaborated with panel data models. THe question I have is: how should I estimate this model? Should I use logarithms to make the functional form linear? If so, should I logarirthm the competitiveness value or the corporate tax rate?