The modified Mellin transform and its inverse on the spaces $LG'_\alpha$, $\alpha>-1$, as well as the modified Mellin convolution and its properties over these spaces are investigated. The spaces $LG'_\alpha$, $\alpha>-1$, are inspected through the spaces of Newton's series using an isomorphism between them. Remarks are given on the domain of convergence of some Dirichlet's series. Finally, the modified Mellin convolution is applied in solving an integro-differential equation.