<p>This study generalizes the classical Taylor-Sedov framework to analyze finite-source spherical blast waves propagating through both uniform and power-law density media. Previous analyses have predominantly focused on the effects of varying initial conditions on blast dynamics. In contrast, this study investigates the primary shock wave evolution within different ambient gases, demonstrating the critical dependence on the initial density ratio between the blast sphere and the ambient medium, as well as the ambient density profile. We derive new scaling laws based on the density ratio, which accurately predict the dimensionless main shock distance. Furthermore, we systematically examine, for the first time, the conditions for uniform volume expansion, uniform surface area growth, and uniform shock wave propagation in power-law density media, revealing a key scaling relation associated with the power-law exponent. Numerical simulations validate these novel theoretical predictions, demonstrating excellent agreement with the normalized solutions. These findings provide new insights into blast wave dynamics in inhomogeneous media and have implications for astrophysical and laboratory plasma environments.</p>