  A natural construction of the logarithmic extension of the M(2,p) minimal
models is presented, which generalises our previous model [0708.0802] of
percolation (p=3). Its key aspect is the replacement of the minimal model
irreducible modules by reducible ones obtained by requiring that only one of
the two principal singular vectors of each module vanish. The resulting theory
is then constructed systematically by repeatedly fusing these building block
representations. This generates indecomposable representations of the type
which signify the presence of logarithmic partner fields in the theory. The
basic data characterising these indecomposable modules, the logarithmic
couplings, are computed for many special cases and given a new structural
interpretation. Quite remarkably, a number of them are presented in closed
analytic form (for general p). These are the prime examples of
``gauge-invariant'' data - quantities independent of the ambiguities present in
defining the logarithmic partner fields. Finally, mere global conformal
invariance is shown to enforce strong constraints on the allowed spectrum: It
is not possible to include modules other than those generated by the fusion of
the model's building blocks. This generalises the statement that there cannot
exist two effective central charges in a c=0 model. It also suggests the
existence of a second ``dual'' logarithmic theory for each p. Such dual models
are briefly discussed.
