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We prove that if $0<T_0<T\leq\infty$, $(\mathbf{u},\mathbf{b},p)$ is asuitable weak solution of the MHD equations in $\mathbb{R}^3\times(0,T)$ andeither $\mathcal{F}_{\gamma}(p_-)\in L^{\infty}(0,T_0;\,L^{3/2}(\mathbb{R}^3))$ or $\mathcal{F}_{\gamma}((|\mathbf{u}|^2+|\mathbf{b}|^2+2p)_+)\in L^{\infty}(0,T_0;\, L^{3/2}(\mathbb{R}^3))$ for some$\gamma>0$, where $\mathcal{F}_{\gamma}(s)=s\, [\ln{}(1+ s)]^{1+\gamma}$ andthe subscripts "$-$" and "$+$" denote the negative and the nonnegative part,respectively, then the solution $(\mathbf{u},\mathbf{b},p)$ has no singularpoints in $\mathbb{R}^3\times(0,T_0]$.

New Regularity Criteria for Weak Solutions to the MHD Equations in Terms of an Associated Pressure