  We investigate the quintessential inflation in the logarithmic Cartan $F(R)$
gravity. A small logarithmic modification of the general relativity has the
potential to introduce both inflation and dark energy. We evaluate the time
evolution of the Universe such as inflation, reheating, and dark energy. The
parameters in the model are fixed to introduce the inflation and the dark
energy scales. We show that the CMB fluctuations induced by the inflation are
consistent with the current observations. In the reheating process, it is
possible to achieve the reheating temperature required for nucleosynthesis in
Big Bang scenario. It can be seen that by choosing an appropriate value for the
scalaron field after reheating, the scalaron field again dominates the energy
of the Universe and causes the current accelerating expansion as dark energy.
