= Solution
Let
$$
e=(1,\ldots,1,2,\ldots,2)\in\mathbb R^{r+s},
$$
with $r$ entries equal to one and $s$ entries equal to two. Its coordinate sum is
$$
r+2s=[K:\mathbb Q]\ne0,
$$
so $e\notin H$. Since the fundamental-unit logarithms form a basis of the codimension-one hyperplane $H$, the vectors
$$
e,\operatorname{Log}(u_1),\ldots,\operatorname{Log}(u_m)
$$
form a basis of $\mathbb R^{r+s}$. This gives unique real numbers $t,\lambda_i$ such that
$$
\operatorname{Log}(\beta)
=te+\sum_{i=1}^m\lambda_i\operatorname{Log}(u_i).
$$
The sum of the coordinates on the left is
$$
\log|N_{K/\mathbb Q}(\beta)|=\log N((\beta))=\log N(I).
$$
Every unit logarithm has coordinate sum zero, so summing the right side gives $t(r+2s)$. Therefore
$$
\boxed{t=\frac{\log N(I)}{[K:\mathbb Q]}}.
$$
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