Derived logarithmic deformation theory and moduli stacks of derived logarithmic structures

This paper investigates the derived and spectral analogs of logarithmic geometry. We develop the deformation theory for animated log rings and ${\mathbb{E}}_{\mathrm{\infty }}$-log rings and examine the corresponding theories of derived and spectral log stacks. Furthermore, we define moduli stacks for derived and spectral log structures and establish their representability. As an application, we will construct $\mathrm{\infty }$-root stacks in the derived and spectral settings and study the associated geometric properties.