The nonproportional scintillation response of LaBr(3) doped with 5% Ce(3+) and of LaCl(3) doped with 10% Ce(3+) was measured using highly monochromatic synchrotron irradiation. To estimate the photon response, pulse height spectra at many finely spaced energy values between 9 and 100 keV were measured. The experiment was carried out at the X-1 beamline at the Hamburger Synhrotronstrahlungslabor (HASYLAB) synchrotron radiation facility in Hamburg, Germany. Special attention was paid to the x-ray fluorescence escape peaks as they provide us with additional information about photon response in the range 1.2-14.5 keV for LaBr(3):Ce and 2.0-11.6 keV for LaCl(3):Ce. A rapid variation of the photon response curve is observed near the lanthanum K-electron binding energy for both scintillators. A dense sampling of data was performed around this energy and those data are used to apply a method, which we call K-dip spectroscopy. This method allows us to derive the electron response curves of LaBr(3):Ce and LaCl(3):Ce down to energies as low as 0.1 keV.