The spectral energy distribution of blazars around the synchrotron peak can be well described by the log-parabolic model that has three parameters: peak energy ($E_textrm{p}$), peak luminosity ($L_textrm{p}$) and the curvature parameter ($b$). It has been suggested that $E_textrm{p}$ shows relations with $L_textrm{p}$ and $b$ in several sources, which can be used to constrain the physical properties of the emitting region and/or acceleration processes of the emitting particles. We systematically study the $E_textrm{p}$-$L_textrm{p}$ and $E_textrm{p}$-(1$/b$) relations for 14 BL Lac objects using the 3-25~keV $RXTE$/PCA and 0.3-10~keV $Swift$/XRT data. Most objects (9/14) exhibit positive $E_textrm{p}$-$L_textrm{p}$ correlations, three sources show no correlation, and two sources display negative correlations. In addition, most targets (7/14) present no correlation between $E_textrm{p}$ and 1$/b$, five sources pose negative correlations, and two sources demonstrate positive correlations. 1ES~1959+650 displays two different $E_textrm{p}$-$L_textrm{p}$ relations in 2002 and 2016. We also analyze $E_textrm{p}$-$L_textrm{p}$ and $E_textrm{p}$-(1$/b$) relations during flares lasting for several days. The $E_textrm{p}$-$L_textrm{p}$ relation does not exhibit significant differences between flares, while the $E_textrm{p}$-(1$/b$) relation varies from flare to flare. For the total sample, when $L_textrm{p}$ < $textrm{10}^textrm{45} textrm{erg} textrm{s}^textrm{-1}$, there seems to be a positive $E_textrm{p}$-$L_textrm{p}$ correlation. $L_textrm{p}$ and the slope of $E_textrm{p}$-$L_textrm{p}$ relation present an anti-correlation, which indicates that the causes of spectral variations might be different between luminous and faint sources. $E_textrm{p}$ shows a positive correlation with the black hole mass. We discuss the implications of these results.
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