  The logarithmic $R^2$-corrected $F(R)$ gravity is investigated as a prototype
model of modified gravity theories with quantum corrections. By using the
auxiliary field method, the model is described by the general relativity with a
scalaron field. The scalaron field can be identified as an inflaton at the
primordial inflation era. It is also one of the dark matter candidates in the
dark energy era. It is found that a wide range of the parameters is consistent
with the current observations of CMB fluctuations, dark energy and dark matter.
