Objective Bayesian inference for the Exponential-Logarithmic distribution

{Posterior propriety conditions are proved and validated by simulation; significance: objective Bayesian analysis of the Exponential–Logarithmic model.} This study explores an objective Bayesian inference approach for parameter estimation in the Exponential-Logarithmic (EL) distribution. Initially, we establish the necessary and sufficient conditions under which improper priors yield proper posterior distributions for the EL distribution. Additionally, we provide sufficient conditions to ensure the finiteness of posterior moments. These theoretical results are specifically applied to Jeffreys' prior, the maximal data information prior, and reference priors, demonstrating that such improper priors indeed generate proper posterior distributions. To assess the impact of the proposed priors on posterior estimation, we employ Markov Chain Monte Carlo methods and conduct extensive numerical simulations, comparing Bayesian estimators with the maximum likelihood estimators in terms of bias, mean squared error, and coverage probability. Lastly, we analyse a real dataset to illustrate the practical applicability of the proposed methodology.