  In this paper, we investigate the homogeneous and isotropic flat FLRW
Universe in the logarithmic form of $f\left( Q\right) $ gravity, where $% Q $
is the non-metricity scalar, specifically, $f\left( Q\right) =\alpha +\beta
\log \left( Q\right) $, where $\alpha $ and $\beta $ are free model parameters.
In this study, we consider a parametrization of the Hubble parameter as
$H\left( z\right) =\eta \left[ (z+1)^{-\gamma }+1\right] $, where $\gamma $ and
$\eta $ are model/free parameters which are constrained by an $R^{2}$-test from
57 points of the Hubble datasets in the redshift range $0.07<z<2.36$. Further,
we investigate the physical properties of the model. We analyze the energy
conditions to check the compatibility of the model. We found the SEC is
violated for the logarithmic form of $f\left( Q\right) $ gravity due to the
reality that the Universe in an accelerating phase. Finally, we discuss some
important cosmological parameters in this context to compare our model with
dark energy models such as jerk parameter and statefinder parameters.
