We consider the asymptotic minimum rate under the logarithmic loss distortion constraint. More specifically, we find the asymptotic minimum rate expression when given distortions get close to 0. The problem under consideration is separate encoding and joint decoding of correlated two information sources, subject to a logarithmic loss distortion constraint. We introduce a test channel, whose transition probability (conditional probability mass function) captures the encoding and decoding process. Firstly, we find the expression for the special case of doubly symmetric binary sources with binary-output test channels. Then the result is extended to the case where the test channels are arbitrary. When given distortions get close to 0, the asymptotic rate coincides with that for the aforementioned special case. Finally, we consider the general case and show that the key findings for the special case continue to hold. / Thesis / Master of Applied Science (MASc)