The logarithmic part of the Lamb shift, the contribution of the relative order α³ log (1/α) to the atomic state energy, is related to the usual infrared divergence. This fact allows one to calculate easily such corrections in positronium, and derive the recoil and electron-electron terms in the Lambshift Hamiltonian in many-electron atoms. Logarithmic energy corrections of the next order, α⁴ log (1/α), are of a different, relativistic nature. Their calculation is reduced to the ordinary perturbation theory for the nonrelativistic Schrödinger equation. The perturbation operators have the Breit-type structure and are found by the calculation of on-mass-shell diagrams. For positronium, the calculated log-arithmic correction survives only in n³S₁ states and constitutes (5/24)mα⁶ log (1/α)/n³. Logarithmic corrections of the relative order α² log (1/α) to the positronium decay rate are also of the relativistic origin and can be easily computed within the same approach. Arguments are presented in favour of a large numerical factor in the (α/π)² correction to the positronium decay rate.