In　this　paper,　we　constructively　solve　the　Riemann　problem　for　the　relativistic　Euler　equations　with　the　logarithmic　equation　of　state.　The　concentration　and　cavitation　phenomena　are　observed　and　analyzed　during　the　process　of　vanishing　pressure　in　the　Riemann　solutions.　It　is　rigorously　proved　that,　as　the　pressure　vanishes,　they　tend　to　the　two　kinds　of　Riemann　solutions　to　the　zero-pressure　relativistic　Euler　equations,　which　include　a　delta　shock　formed　by　a　weighted　d-measure　and　a　vacuum　state.