  In this paper, we study the constraints on the $f(Q)=Q/(8\pi G) - \alpha
\ln(Q/Q_0)$, symmetric teleparallel model using local measurements and
gravitational wave mock standard sirens. Using observational local SNIa and BAO
data and energy conditions, the logarithmic $f(Q)$ model is capable of
explaining the cosmic late-time acceleration by geometrical means. This result
suggests that the logarithmic symmetric teleparallel model could be a candidate
to solve the cosmological constant problem. In the case of the simulated
standard siren data, by using the performance of the future ET and LISA
detectors, we expect to be able to measure the current Hubble constant $H_0$,
and the matter content $\Omega_m$, with a precision better than 1\% and 6\%,
respectively. Furthermore, we explore the predicted $f(Q)$ logarithmic model
deviation from the standard GR using ET and LISA mock standard sirens. The
ratio $d_L^{\text{gw}}(z)/d_L^{\text{em}}(z)$, which quantifies the deviation
from GR gives us a significant deviation higher than 13\% at $z=1$, and it
continues growing to reach a deviation higher than 18\% in its median value.
Future standard siren data will be able to quantify the strength of the
deviation from GR and hence whether a cosmology like the one implied by this
$f(Q)$ model is feasible.
