On the bases of the image of $2$-adic logarithm on the group of principal units

This paper computes the bases of the image of $2$-adic logarithm on the group of the principal units in all 7 quadratic extensions of ${\mathbb{Q}}_2$. This helps one to understand the free module structure of $2$-adic logarithm at arbitrary points on its domain. We discuss some applications at the end.