We consider the following viscoelastic equation involving variable exponent nonlinearities: u tt-∆ u+∫ 0 tg(t-s)∆u(s)ds+a|u t| m(x)-2u t=|u| q(x)-2u. Generally speaking, the presence of the source drives the system towards instability while the appearance of damping makes the system stable, which makes the interplay between damping and nonlinearity more delicate. As far as we know, when one tries to apply multiplier method, the usual strategy for dealing with the nonlinear term ∫ Ω|u t| m-2u tudx is to apply Young’s inequality and Sobolev embedding inequality to guarantee that this moment will be controlled by both energy functional and its derivative. Therefore, for supercritical damping(m>2N/(N-2)), the failure of the embedding inequality makes the aforementioned technique unsuccessful in our problem. To do this, our strategy is to give a priori estimate for the integral ∫ Ω(2+t) 1-m(x)|u| m(x)dx and then apply modified multiplier method to give the energy functional decays logarithmically. Meanwhile, we also give explicit dependence of decay rate on both the exponent m(x) and the relaxation function function g. These results extend the paper [S. A. Messaoudi, M. M. Al-Gharabli and A. M. Al-Mahdi, Math. Methods Appl. Sci., 45(14)(2022): 8389-8411].