  The complete contribution to the muonium hyperfine splitting of relative
order alpha^3(m_e/m_mu)ln(alpha) is calculated. The result amounts to 0.013
kHZ, much smaller than suggested by a previous estimate, and leads to a 2-sigma
shift of the most precise value for the muon-electron mass ratio, with the
error reduced by approximately 30%. Analogous contributions are calculated for
the positronium hyperfine splitting:
(217/90-17*ln{2}/3)m_e(alpha^7/pi)ln(1/alpha) \approx -0.32 MHz; the remaining
theoretical uncertainty should be well below experimental error, leaving
discrepancies of 2.5-sigma and 3.5-sigma with the two most precise
measurements.
