  In this work we investigate the rate of convergence to equilibrium for
subcritical solutions to the Becker-D\"oring equations with physically relevant
coagulation and fragmentation coefficients and mild assumptions on the given
initial data. Using a discrete version of the log-Sobolev inequality with
weights we show that in the case where the coagulation coefficient grows
linearly and the detailed balance coefficients are of typical form, one can
obtain a linear functional inequality for the dissipation of the relative free
energy. This results in showing Cercignani's conjecture for the Becker-D\"oring
equations and consequently in an exponential rate of convergence to
equilibrium. We also show that for all other typical cases one can obtain an
'almost' Cercignani's conjecture that results in an algebraic rate of
convergence to equilibrium. Additionally, we show that if one assumes an
exponential moment condition one can recover Jabin and Niethammer's rate of
decay to equilibrium, i.e. an exponential to some fractional power of $t$.
