  In an attempt to look for the root of nonstandard Lagrangians in the theories
of the inverse variational problem we introduce a logarithmic Lagrangian (LL)
in addition to the so-called reciprocal Lagrangian (RL) that exists in the
literature. The equation of motion that follows from the RL could easily be
solved by using its first or Jacobi integral. This is not, however, true for
the similar equation resulting from the LL. We make use of a method of
factorization to find the particular solutions for equations following from
both RL an LL and subsequently derive a novel approach to obtain their general
solutions from the particular ones. The case studies presented by us include
the modified Emden-type equation for which we also construct a new expression
for the Lagrangian function (NL). We point out that NL is not related to RL by
a total derivative or gauge term. In spite of that NL leads to the same
equation of motion as that given by the RL.
