A logging tool is for determining a property of a subsurface formation having a borehole therein and includes a housing to fit within the borehole. An x-ray generator is carried by the housing and includes at least three extractor electrodes, an electron emitter to emit electrons toward the extractor electrodes, and a target downstream of the extractor electrodes. The extractor electrodes have independently selectable potentials so as to allow direction of an electron beam, formed from the electrons emitted by the electron emitter, toward different longitudinal and lateral regions of the target, the target to emit x-rays into the subsurface formation when struck by the electron beam. A radiation detector is carried by the housing to detect incoming radiation resulting from interactions between the x-rays and the subsurface formation. Processing circuitry is coupled to the radiation detector to determine the property of the subsurface formation based upon detected incoming radiation.