  In this paper, we study the dynamics of non-interacting and interacting
holographic dark energy models in the framework of Brans-Dicke theory. As
system's infra-red cut-off we consider the future event horizon. The motivation
of this work is to use the logarithmic form of the Brans-Dicke scalar field,
$\phi \propto ln(\alpha+\beta a)$, where $\alpha$ and $\beta$ are constants and
`a' is the scalar factor as proposed Kumar and Singh in a recent work to study
the new agegraphic dark energy models. We find the time-dependent equation of
state parameter and deceleration parameter which describe the phase transition
of the universe. We observe that the model explains the early time inflation
and late time acceleration including matter-dominated phase. It is also
observed that the equation of state parameter may cross phantom divide line in
late time evolution. The cosmic coincidence problem is also discussed for both
the models. We observe that this logarithmic form of Brans-Dicke scalar field
is more appropriate to achieve a less acute coincidence problem in
non-interacting model whereas a soft coincidence can be achieved if coupling
parameter in interacting model has small value.
