We show that the (p, p') Clarkson's inequality holds in the Edmunds-Triebel logarithmic spaces Aq(logA)b,q and in the Zygmund spaces Lp(logL)b(W), for b Î \mathbbR and for suitable 1 £ p £ 2. As a consequence of these results we also obtain some new information about the types and the cotypes of these spaces.