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We present a tunable continuous-wave terahertz radiation source based on photomixing of a tunable dual-mode semiconductor laser on a log-periodic antenna. The discontinuously tunable dual-mode emission of the laser is achieved by spectrally filtered feedback in a double-external-cavity configuration. Two different two-frequency lasing regimes have been identified: pure dual-mode and dual-mode-comb emission. We spectrally investigate the laser emission in these two regimes and study the consequences for the generation of terahertz radiation. Under pure dual-mode emission, we achieve higher power, higher coherence and good tunability of the THz radiation. We demonstrate by detecting absorption lines of water vapor and hydrochloric acid gas that our concept for a tunable continuous-wave terahertz radiation source offers attractive properties for spectroscopic applications. [ABSTRACT FROM AUTHOR]