A method for determining at least one electrical property of an earth formation includes emitting an electromagnetic signal into the earth formation from an antenna and measuring an electromagnetic signal from the earth formation. The antenna is a broadband log antenna mounted on a substrate having at least a high dielectric permittivity, defined as a dielectric permittivity of about ∈=100 to ∈=1000 or a gigantic dielectric permittivity, defined as a dielectric permittivity of about ∈=1000 or greater. The antenna has a radius between about 2.5 millimeters (mm) and 10 centimeters (cm). The method further includes determining at least one electrical property of one or more of a borehole, a borehole fluid, and the earth formation based on measuring the electromagnetic signal.