The logarithmic Kodaira dimension introduced by S. Iitaka [1] plays an important role in the study of non-compact algebraic varieties. Especially it seems interesting to find the logarithmic Kodaira dimension ĸ^^-, ( P^2 -C), where C is an algebraic curve in P^2(C) with degree n. But it is difficult in the case when C is rational and has only one singular point which is a cusp, because we do not know whether such curves exist for n≧6. We shall prove here the non-existence of such curves under the condition that n=6 and the multiplicity of the cusp is 2.