  In this paper, an interacting holographic dark energy model with Hubble
horizon as an infra-red cut-off is considered in the framework of Brans-Dicke
theory. We propose a logarithmic form $\phi \propto ln(\alpha+\beta a)$ of the
Brans-Dicke scalar field to alleviate the problems of interacting holographic
dark energy models in Brans-Dicke theory. We find that the equation of state
parameter $w_h$ and deceleration parameter $q$ are negative in the early time
which shows the early time inflation. During the evolution the sign of
parameter $q$ changes from negative to positive which means that the Universe
expands with decelerated rate whereas the sign of $w_h$ may change or remain
negative throughout the evolution depending on the values of parameters. It is
also observed that $w_h$ may cross the phantom divide line in the late time
evolution. The sign of $q$ changes from positive to negative during late time
of evolution which explains the late time accelerated expansion of the
Universe. Thus, we present a unified model of holographic dark energy which
explains the early time acceleration (inflation), medieval time deceleration
and late time acceleration. We also discuss the cosmic coincidence problem. We
obtain a time-varying density ratio of holographic dark energy to dark matter
which is a constant of order one ($r\sim \mathcal{O}(1)$) during early and late
time evolution. Therefore, our model successfully resolves the cosmic
coincidence problem.
