Logarithm isomorphism on deep principal units ($\log :1+{\pi }^r{\mathcal{O}}_K\cong {\pi }^r{\mathcal{O}}_K,r>v_K(\ell )/(\ell -1)$)

For a mixed-characteristic local field with residue characteristic $\ell$, the p-adic logarithm and p-adic exponential function are mutually inverse continuous group homomorphisms on these domains. Bounds on the valuations of the denominators show that higher terms have greater valuation than the linear term. Dividing the logarithm by ${\pi }^r$ identifies the multiplicative higher principal-unit group with the additive integer ring; in particular it is torsion-free.