We describe the application of a perfectly matched layer (PML) combined with a self-adaptive goal-oriented $hp$-finite element (FE) method to the simulation of borehole resistivity measurements. The adaptive refinements and fast convergence of the self-adaptive $hp$-FE method enhance the performance of the PML, thereby enabling the accurate and efficient truncation of the computational domain in open-domain problems. We apply this method to the simulation of axisymmetric through-casing resistivity measurements acquired in a borehole environment that are typically used for the assessment of rock formation properties. Our numerical results confirm the accuracy and efficiency of our method and provide evidence of highly accurate and reliable simulations of borehole logging measurements in the presence of a conductive steel casing and material contrast of fourteen orders of magnitude in electrical conductivity. Moreover, the combination of adaptivity and PML enables us to significantly reduce the size of the computational domain with substantial savings in computer time and memory.