  A unification of dark matter and dark energy in terms of a logotropic perfect
dark fluid has recently been proposed, where deviations with respect to the
standard $\Lambda {\rm CDM}$ model are dependent on a single parameter $B$. In
this paper we show that the requirement that the linear growth of cosmic
structures on comoving scales larger than $8 h^{-1} \, {\rm Mpc}$ is not
significantly affected with respect to the standard $\Lambda {\rm CDM}$ result
provides the strongest constraint to date on the model ($B <6 \times 10^{-7}$),
an improvement of more than three orders of magnitude over previous constraints
on the value of $B$. We further show that this constraint rules out the
logotropic Unified Dark Energy model as a possible solution to the small scale
problems of the $\Lambda$CDM model, including the cusp problem of Dark Matter
halos or the missing satellite problem, as well as the original version of the
model where the Planck energy density was taken as one of the two parameters
characterizing the logotropic dark fluid.
