  The modified $F(R)$ gravity theory with the function
$F(R)=-(1/\beta)\ln(1-\beta R)$ is studied. The action at small coupling
$\beta$ becomes Einstein--Hilbert action. The bound on the parameter $\beta$
from local tests is $\beta\leq 2\times 10^{-6}$ cm$^2$. We find the constant
curvature solutions and it was shown that the de Sitter space is unstable but a
solution with zero Ricci scalar is stable. The potential and the mass of the
scalar field (scalaron) are obtained in the Einstein's frame. The slow-roll
cosmological parameters are studied and e-folds number is evaluated. The
critical points of autonomous equations are analyzed. The function $m(r)$ that
describes the deviation from the $\Lambda$CDM model is calculated.
