Let B denote the unit ball in $C^n$, and ν the normalized Lebesgue measure on B. For $alpha$ > -1, define $dv_alpha$(z) ＝ $c_alpha$$(1-midzmid^2)^{alpha}$dν(z), z $in$ B. Here $c_alpha$ is a positive constant such that $v_alpha$(B) ＝ 1. Let H(B) denote the space of all holomorphic functions in B. For $pgeq1$, define the Bergman-Privalov space $(AN)^{p}(v_alpha)$ by $(AN)^{p}(v_alpha)$ = ${finH(B)$ : $int_B{log(1+midfmid)}^pdv_alpha;<;infty}$ In this paper we prove that a function $finH(B)$ is in $(AN)^{p}$$(v_alpha)$ if and only if $(1+midfmid)^{-2}{log(1+midfmid)}^{p-2}mid ablafmid^2;epsilon;L^1(v_alpha)$ in the case 1＜p＜$infty$, or $(1+midfmid)^{-2}midfmid^{-1}mid{ abla}fmid^2;epsilon;L^1(v_alpha)$ in the case p ＝ 1, where $nabla$f is the gradient of f with respect to the Bergman metric on B. This is an analogous result to the characterization of the Hardy spaces by M. Stoll [18] and that of the Bergman spaces by C. Ouyang-W. Yang-R. Zhao [13].