We propose a phenomenological study of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach applied to the data on the proton structure function $F_2$ measured at HERA in the small-$x_{Bj}$ region ($x_{Bj}<0.01$) and $Q^2$ in the range 5 to 120 GeV$^2$. With a simplified ``effective kernel\'\' approximation, we present a comparison between leading-logs (LO) and next-to-leading logs (NLO) BFKL approaches in the saddle-point approximation, using known resummed NLO-BFKL kernels. The LO result gives a very good description of the data with a three parameters fit of $F_2$ but an unphysical value of the strong coupling constant, whereas the NLO two parameters fit leads to a qualitatively satisfactory account of the running coupling constant effect but quantitatively, for $Q^2 < 10$ GeV$^2$, it fails to reproduce properly the data.
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