Using a sample of $4.48times10^{8}$ $psi(3686)$ events collected with the BESIII detector at the BEPCII collider, we study the two-photon decays of the pseudoscalar mesons $pi^0$, $eta$, $eta^prime$, $eta(1405)$, $eta(1475)$, $eta(1760)$, and $X(1835)$ in $J/psi$ radiative decays using $psi(3686)topi^{+}pi^{-}J/psi$ events. The $pi^0$, $eta$ and $eta^prime$ mesons are clearly observed in the two-photon mass spectra, and the branching fractions are determined to be $B(J/psitogammapi^{0}to3gamma)=(3.57pm0.12pm0.16)times10^{-5}$, $B(J/psitogammaetato3gamma)=(4.42pm0.04pm0.18)times10^{-4}$, and $B(J/psitogammaetato3gamma)=(1.26pm0.02pm0.05)times10^{-4}$, where the first errors are statistical and the second systematic. No clear signal for $eta(1405)$, $eta(1475)$, $eta(1760)$ or $X(1835)$ is observed in the two-photon mass spectra, and upper limits at the $90%$ confidence level on the product branching fractions are obtained.