  We initiate the study of deformation theory in the context of derived and
higher log geometry. After reconceptualizing the "exactification"-procedures in
ordinary log geometry in terms of Quillen's approach to the cotangent complex,
we construct an "exactified tangent bundle" over the category of log ring
spectra. The fibers recover the categories of modules over the underlying ring
spectra, and the resulting cotangent complex functor specializes to log
topological Andr\'e--Quillen homology on each fiber. As applications, we
characterize log square-zero extensions and derive a log variant of \'etale
rigidity, applicable to some tamely ramified extensions of ring spectra.
