Roughness measures for uncertainty data occur with less consideration since the data involve indeterminacy and inconsistency. The indeterminacy plus inconsistency can be solved by a rough neutrosophic set with roughness approximation. Therefore, a binary logarithm similarity measure for a rough neutrosophic set with roughness approximation was proposed in this research. A rough neutrosophic set was chosen as the uncertainty set theory information, which includes the upper and lower approximation with a boundary set approximation. The objectives of this research are to define a binary logarithm similarity measure for a rough neutrosophic set, to formulate the properties satisfied by the proposed similarity measure, and to develop a decision-making model by using a binary logarithm similarity measure for a case study (COVID-19). The roughness approximation was used in the derivation of the binary logarithm similarity measure. The proving result was finalized. Then, the derivation of binary logarithm similarity measures of a rough neutrosophic set was well defined. As a validation process, the similarity properties for identifying the most important priority group for COVID-19 vaccines were used such as age, health state, women, and job types. Following that, the decision-making for identifying the most important priority group for COVID-19 vaccines is presented.