Equivalence of Definitions of P-adic Valuation on P-adic Numbers

Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Proof
Let $x \in \Q_p \setminus \set 0$.

Let $l$ be the index of the first non-zero coefficient in the $p$-adic expansion:
 * $l = \min \set {i: i \ge m \land d_i \ne 0}$

From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:
 * $\norm x_p = p^{-l}$

By definition of real general logarithm:
 * $-\log_p \norm x_p = l$

The result follows.