A logging system for measuring anisotrophic properties of the materials penetrated by a borehole. A downhole or “logging tool” element of the system comprises a source section that comprises either a unipole or a dipole acoustic source. The receiver section comprises a plurality of receiver stations disposed at different axial spacings from the acoustic source. Each receiver station comprises one or more acoustic receivers. The system requires that the source and receiver sections rotate synchronously as the logging tool is conveyed along the borehole. Receiver responses are measured in a plurality of azimuthal angle segments and processed as a function of rotation angle of the tool. The logging system can be embodied as a logging-while-drilling system, a measurement-while-drilling system, and a wireline system that synchronously rotates source and receiver sections. All embodiments require that the acoustic source operate at a relatively high frequency.