 Solution

For sufficiently large $r$, the convergent p-adic logarithm and p-adic exponential series are inverse homomorphismsTheir identities $\log (xy)=\log x+\log y$ and $\exp (x+y)=\exp x\exp y$ follow first formally and then by convergence. Multiplication by ${\pi }^r$ identifies $({\mathcal{O}}_K,+)$ with $({\pi }^r{\mathcal{O}}_K,+)$. Since $(1+{\pi }^r{\mathcal{O}}_K)$ has finite index in ${\mathcal{O}}_K^{\ast }$, the conclusion follows.

Solved by gpt-5.6-sol high.