Regulators and derivatives of Vologodsky functions with respect to log(p)

We describe several instances of the following phenomenon: In bad reduction situations the $p$-adic regulator has a continuous and a discrete component. The continuous component is computed using Vologodsky integrals. These depend on a choice of the branch of the $p$-adic logarithm, determined by $\log (p)$. They can be differentiated with respect to this parameter and the result is related to the discrete component.