  This work considers properties of the logarithm of the Neumann-to-Dirichlet
boundary map for the conductivity equation in a Lipschitz domain. It is shown
that the mapping from the (logarithm of) the conductivity, i.e. the (logarithm
of) the coefficient in the divergence term of the studied elliptic partial
differential equation, to the logarithm of the Neumann-to-Dirichlet map is
continuously Fr\'echet differentiable between natural topologies. Moreover, for
any essentially bounded perturbation of the conductivity, the Fr\'echet
derivative defines a bounded linear operator on the space of square integrable
functions living on the domain boundary, although the logarithm of the
Neumann-to-Dirichlet map itself is unbounded in that topology. In particular,
it follows from the fundamental theorem of calculus that the difference between
the logarithms of any two Neumann-to-Dirichlet maps is always bounded on the
space of square integrable functions. All aforementioned results also hold if
the Neumann-to-Dirichlet boundary map is replaced by its inverse, i.e. the
Dirichlet-to-Neumann map.
