p-adic exponential function (${\exp }_p$)

The $p$-adic exponential $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ converges near zero.

On ${\mathbb{C}}_p$, the power series $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ has radius of convergenceIndeed, Legendre formula gives $v_p(n!)/n\to 1/(p-1)$, and the Cauchy-Hadamard theorem gives the stated radius.

The $p$-adic logarithm is the convergent series $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$ on a sufficiently small group of principal units.
It is locally inverse to the p-adic exponential function.

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.