A numerically efficient transfer matrix (TM) approach is introduced to investigate the long-range Ising spin chain. Results obtained within this procedure are primarily used to verify the Tsallis scaling hypothesis for long-range systems with an alpha power-law decay of the interaction constants, both in the extensive (alpha>1) and nonextensive (alpha<1) regimes. Results for finite-size systems, taking into account all interactions between spins up to 24 sites apart, show that the conjecture is satisfied with a very good precision (less than 0.004%) for all temperature intervals. This TM procedure is further used to investigate several other thermodynamic and critical properties of this system, and it may also be extended to similar one-dimensional long-range systems.