AK vikings advice about building on PT base is spot on, additionaly, debark the logs if you want longevity.
I know everyone want to see logs on the interior, but if you observe Scandinavian and Russian log construction (the experts and originators of log construction), you will see that most of their interior walls are furred out, often with insulation between the logs and the interior surface. This is something to consider.
Probably not necessary for a play house or storage or whatever after she outgrows it. But I've stayed in log cabins/houses that are just logs, and ones that are furred out and insulated, the difference in both summer and winter is unbelievable. Logs are just not very good insulation. They have a reputation for being good insulation, but I believe that is because back in the old days they were comparing them to board houses with nothing, or at most a simple air space between walls as insulation.
Also, rounded logs on the inside tend to collect dust, spider webs, etc. and are a pain to clean. And not nearly as easy to attach things, from hanging pictures, to cabinets. If you aren't going to cover the logs inside, I would recommend squaring them off on the inside, to facilitate cleaning and interior decorating/modifying. Again, possibly not necessary for a play house, but something to think about if you plan to keep and use it.