We study the estimation of Tsallis entropy of a finite number of independent populations, each following an exponential distribution with the same scale parameter and distinct location parameters for $q>0$. We derive a Stein-type improved estimate, establishing the inadmissibility of the best affine equivariant estimate of the parameter function. A class of smooth estimates utilizing the Brewster technique is obtained, resulting in a significant improvement in the risk value. We computed the Brewster-Zidek estimates for both one and two populations, to illustrate the comparison with best affine equivariant and Stein-type estimates. We further derive that the Bayesian estimate, employing an inverse gamma prior, which takes the best affine equivariant estimate as a particular case. We provide a numerical illustration utilizing simulated samples for a single population. The purpose is to demonstrate the impact of sample size, location parameter, and entropic index on the estimates.