There are simple equations relating Rényi entropy and q-log entropy if we measure both in nats:
I mentioned in the post on q-log entropy that there were two possible ways it could be defined. The equation above applies to what I called in that post Sq, not S ‘q. In other words, it applies to the version that uses lnq(1/p) and not the version that uses -lnq(p). Recall that these are not equal unless q equals 1. When q does equal 1, then Rényi entropy and q-log entropy are the same as Shannon entropy.
Source: Tom Leinster, Entropy and Diversity: The Axiomatic Approach.