  Penguin diagrams for decays such as $b\to (s,d)\gamma$ involve virtual loops
of $u$ or other light quarks. Logarithms of the virtual quark mass could, in
principle, influence the phenomenological analysis of the decay. It is thus
important to study these logarithms to all orders in QCD perturbation theory.
In this paper we show that, at arbitrary order, the matrix elements of
operators in the effective hamiltonian contributing to $b\to s\gamma$ are
finite for the limit of $m_u \to 0$ in penguin loops.
