We report the measurement of a neutron radiative cross section on a target with a high gamma activity, at the Los Alamos DANCE detector (Detector for Advanced Neutron Capture Experiments) for the first time. The (n,γ) reaction properties of the unstable Lu173 isotope were studied. Two experimental campaigns were needed to determine the Lu173(n,γ) cross section. They were performed at the Los Alamos Neutron Science Center (LANSCE) spallation neutron source facility. To this end, two targets were produced from an Hf sample using successively proton irradiation and chemical separations. They were composed of Lu173,Lu174, and Lu175 isotopes. The two measurements were conducted at two-year intervals, taking advantage of the difference of the isotopic decay lifetimes in order to identify resonances from the Lu173(n,γ) and Lu174(n,γ) reactions. Just over one hundred new resonances were observed, the majority of which come from the Lu173(n,γ) reaction. Only a few of them were assigned to the Lu174(n,γ) reaction. As regards the Lu173(n,γ) reaction, the radiative neutron capture cross section was determined over the energy range from thermal neutron up to 200 eV. The parameters of resonances in the resolved resonance region were extracted with the sammy code while calculations with talys determined the capture cross section in the unresolved resonances region. At the keV region, we performed standard talys calculations as well as other microscopic investigations by substituting the standard talys photon strength function by another one from QRPA models.