We discuss the present situation in precision experimental study of elastic electron-nucleon scattering, which give information about effects beyond the one-photon exchange. Such a study is important to get reliable information on the nucleon structure at short distances. Special interest is connected with study of one-particle polarization observables, the beam and target spin asymmetry. This quantities disappear in the one-photon exchange and thus give a direct information on the two-photon exchange in electron-nucleon scattering. We give predictions for the target asymmetry and compare our theoretical results for the beam asymmetry with experiment in the resonance region (electron laboratory energy up to few GeV). We discuss role of large logarithms, $|n^2(Q^2/m^2)$ and $|n(Q^2/m^2)$ (where $m$ is the electron mass and -$Q^2$ is the momentum transfer squared), which enhance the beam spin asymmetry.