We report on a measurement of the cosmic ray composition by the Telescope Array Low-Energy Extension (TALE) air fluorescence detector (FD). By making use of the Cherenkov light signal in addition to air fluorescence light from cosmic ray (CR) induced extensive air showers, the TALE FD can measure the properties of the cosmic rays with energies as low as $sim 2$ PeV and exceeding 1 EeV. In this paper, we present results on the measurement of $X_{rm max}$ distributions of showers observed over this energy range. Data collected over a period of $sim 4$ years was analyzed for this study. The resulting $X_{rm max}$ distributions are compared to the Monte Carlo (MC) simulated data distributions for primary cosmic rays with varying composition and a 4-component fit is performed. The comparison and fit are performed for energy bins, of width 0.1 or 0.2 in $log_{10} (E/{rm eV})$, spanning the full range of the measured energies. We also examine the mean $X_{rm max}$ value as a function of energy for cosmic rays with energies greater than $10^{15.8}$ eV. Below $10^{17.3}$ eV, the slope of the mean $X_{rm max}$ as a function of energy (the elongation rate) for the data is significantly smaller than that of all elements in the models, indicating that the composition is becoming heavier with energy in this energy range. This is consistent with a rigidity-dependent cutoff of events from galactic sources. Finally, an increase in the $X_{rm max}$ elongation rate is observed at energies just above $10^{17}$ eV indicating another change in the cosmic rays composition.
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