  In various contexts in mathematical physics one needs to compute the
logarithm of a positive unbounded operator. Examples include the von Neumann
entropy of a density matrix and the flow of operators with the modular
Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation
where the operator under consideration, that we denote by $\Delta$, can be
related by a perturbative series to another operator $\Delta_0$, whose
logarithm is known. We set up a perturbation theory for the logarithm $\log
\Delta$. It turns out that the terms in the series possess remarkable algebraic
structure, which enable us to write them in the form of nested commutators plus
some "contact terms."
