  We investigate the exclusive semileptonic $B_c\to
(D,\eta_c,B,B_s)\ell\nu_\ell$, $\eta_b\to B_c\ell\nu_\ell$($\ell=e,\mu,\tau$)
decays using the light-front quark model constrained by the variational
principle for the QCD motivated effective Hamiltonian. The form factors
$f_+(q^2)$ and $f_-(q^2)$ are obtained from the analytic continuation method in
the $q^+=0$ frame. While the form factor $f_+(q^2)$ is free from the zero-mode,
the form factor $f_-(q^2)$ is not free from the zero-mode in the $q^+=0$ frame.
We quantify the zero-mode contributions to $f_-(q^2)$ for various semileptonic
$B_c$ decays. Using our effective method to relate the non-wave function vertex
to the light-front valence wave function, we incorporate the zero-mode
contribution as a convolution of zero-mode operator with the initial and final
state wave functions. Our results are then compared to the available
experimental data and the results from other theoretical approaches. Since the
prediction on the magnetic dipole $B^*_c\to B_c+\gamma$ decay turns out to be
very sensitive to the mass difference between $B^*_c$ and $B_c$ mesons, the
decay width $\Gamma(B^*_c \to B_c \gamma)$ may help in determining the mass of
$B^*_c$ experimentally. Furthermore, we compare the results from the harmonic
oscillator potential and the linear potential and identify the decay processes
that are sensitive to the choice of confining potential. From the future
experimental data on these sensitive processes, one may obtain more realistic
information on the potential between quark and antiquark in the heavy meson
system.
