<p style='text-indent:20px;'>In this paper we consider the stochastic <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional (<inline-formula><tex-math id="M2">\begin{document}$ d = 2, 3 $\end{document}</tex-math></inline-formula>) magneto-hydrodynamic (MHD) equations with fractional dissipations <inline-formula><tex-math id="M3">\begin{document}$ ((-\Delta)^\alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^\beta) $\end{document}</tex-math></inline-formula> driven by degenerate noise. Here <inline-formula><tex-math id="M5">\begin{document}$ \alpha, \beta $\end{document}</tex-math></inline-formula> represent the parameters of the fractional dissipations corresponding to the velocity field and that to the magnetic field respectively. Using the asymptotic coupling method, the asymptotic log-Harnack inequality is established when <inline-formula><tex-math id="M6">\begin{document}$ \alpha\wedge\beta\geq\frac{1}{2}+\frac{d}{4} $\end{document}</tex-math></inline-formula>. As applications, we derive the gradient estimate, asymptotic irreducibility, asymptotic heat kernel estimate and ergodicity for the model.</p>