  We construct plane wave backgrounds with time-dependent profiles
corresponding to Penrose limits of NS5-branes with transverse space symmetry
group broken from SO(4) to SO(2)\times Z_N. We identify the corresponding exact
theory as the five-dimensional Logarithmic Conformal Field Theory (CFT) arising
from the contraction of the SU(2)_{N'}/U(1)\times SL(2,R)_{-N} exact CFT, times
R^5. We study several general aspects and construct the free field
representation in this theory. String propagation and spectra are also
considered and explicitly solved in the light-cone gauge.
