A detail on the proof of equivalence of valuations

Question

I have trouble understanding a step of Neukirch's proof of the characterization of equivalent valuations.

The step is

Now let $y \in K$ be a fixed element satistying $|y|_1>1$. Let $x\in K$, $x\neq 0$. Then $|x|_1=|y|_1^\alpha$ for some $\alpha \in \mathbb{R}$.

I'm sure it is something obvious, but I don't see it.

Answer 1

Just take logarithms: $\alpha=\log|x_1|/\log|y_1|$ will do, we can do this because $|x_1|>0$ and $|y_1|>1$.