  Recently, corrections to the standard Einstein-Hilbert action are proposed to
explain the current cosmic acceleration in stead of introducing dark energy. We
discuss the Palatini formulation of the modified gravity with a $\ln R$ term
suggested by Nojiri and Odintsov. We show that in the Palatini formulation, the
$\ln R$ gravity can drive a current exponential accelerated expansion and it
reduces to the standard Friedmann evolution for high redshift region. We also
discuss the equivalent scalar-tensor formulation of the theory. We indicate
that the $\ln R$ gravity may still have a conflict with electron-electron
scattering experiment which stimulates us to pursue a more fundamental theory
which can give the $\ln R$ gravity as an effective theory. Finally, we discuss
a problem faced with the extension of the $\ln R$ gravity by adding $R^m$
terms.
