We demonstrate a method to tune the carrier concentration of a high temperature superconductor over a wide range, using an applied electric field. Thin film devices were made in an electrical double layer transistor configuration utilizing an ionic liquid. In this way, the surface carrier density in La2-xSrxCuO4 films can be varied between 0.01 and 0.14 carriers per Cu atom with a resulting change in critical temperature of 25 K (70% of the maximum critical temperature in this compound). This allows one to study a large segment of the cuprate phase diagram without altering the level of disorder. We used this method [A. T. Bollinger et al., Nature 472, 458–460 (2011)] to study the quantum critical point at the superconductor to insulator phase transition on the underdoped side of superconducting dome, and concluded that this transition is driven by quantum phase fluctuations and Cooper pair delocalization.