A unified approach is presented for establishing
a broad class of Cram\'er-Rao inequalities for the location parameter,
including, as special cases,
the original inequality of Cram\'er and Rao, as well as an $L^p$ version recently
established by the authors. The new approach allows for
generalized moments and Fisher information measures to be defined by convex
functions that are not necessarily homogeneous.
In particular, it is shown that associated with any log-concave random
variable whose density satisfies certain boundary conditions is a
Cram\'er-Rao inequality for which the given log-concave random
variable is the extremal. Applications to specific instances are
also provided.