Inferences about the ratio of two lognormal means δ can depend on plausible values of ρ, the ratio of the normal standard deviations associated to these distributions. This aspect is not usually considered in most of the analyses carried out in some applied sciences. In this paper we propose a profile likelihood approach that allows the comparison of two independent lognormal data sets in a more exhaustive way. Inferences about δ, ρ and (δ, ρ) are jointly analyzed through a simple closed-form expression obtained for the profile likelihood function of the parameter vector (δ, ρ). A similar analysis is done for ψ and ρ, where ψ is the ratio of two lognormal medians, obtaining also a simple closed-form expression for the profile likelihood function of these parameters. These expressions allow us to construct likelihood contour plots that capture most of the information provided by the samples and become valuable to identify if a trade-off between the parameters under study occurs; in case of that, individual inferences should be analyzed carefully. A detailed series of Monte Carlo simulations are included; they illustrate the performance of profile likelihood and parametric bootstrap approaches, for different sample sizes and parameter values.