  We provide a world-sheet interpretation to the plane wave limit of a large
class of exact supergravity backgrounds in terms of logarithmic conformal field
theories. As an illustrative example, we consider the two-dimensional conformal
field theory of the coset model SU(2)_N/U(1) times a free time-like boson
U(1)_{-N}, which admits a space-time interpretation as a three-dimensional
plane wave solution by taking a correlated limit \`a la Penrose. We show that
upon a contraction of Saletan type, in which the parafermions of the compact
coset model are combined with the free time-like boson, one obtains a novel
logarithmic conformal field theory with central charge c=3. Our results are
motivated at the classical level using Poisson brackets of the fields, but they
are also explicitly demonstrated at the quantum level using exact operator
product expansions. We perform several computations in this theory including
the evaluation of the four-point functions involving primary fields and their
logarithmic partners, which are identified. We also employ the extended
conformal symmetries of the model to construct an infinite number of
logarithmic operators. This analysis can be easily generalized to other exact
conformal field theory backgrounds with a plane wave limit in the target space.
