= Solution
A <Qp-Banach algebra> is a $\mathbb Q_p$-algebra with a complete non-Archimedean submultiplicative norm compatible with the <p-adic absolute value>. A p-valued group is <p-saturated> when it is complete and every $g$ with $\omega(g)>p/(p-1)$ has a pth root.
In the completed rational Iwasawa algebra $\widehat{\mathbb Q_pG}$, the filtration of $g-1$ is $\omega(g)>1/(p-1)$. Since $v_p(n)=O(\log n)$, the terms of
$$
\log g=\sum_{n\geq1}\frac{(-1)^{n+1}}n(g-1)^n
$$
have filtrations tending to infinity, so the series converges. The <Baker--Campbell--Hausdorff formula> expresses $\log(gh)$ in terms of $\log g$ and $\log h$, and its commutator expansion expresses $\log([g,h])$ with leading term $[\log g,\log h]$. Completeness and p-divisibility allow the higher terms to be removed successively. Integer powers give $\log(g^n)=n\log g$, and continuity extends scalar multiplication to $\mathbb Z_p$. Thus the <Lazard logarithm of a p-saturated group> shows that $\log(G)$ is a $\mathbb Z_p$-Lie subalgebra.
The group-like coproduct identity $\Delta(g)=g\otimes g$ implies
$$
\Delta(\log g)=\log g\otimes1+1\otimes\log g,
$$
so each logarithm is primitive. If $(g_1,\ldots,g_d)$ is an ordered basis, their initial forms are linearly independent. Conversely, for a primitive element, its lowest initial form must be linear rather than a product; subtracting a $\mathbb Q_p$-linear combination of the $\log(g_i)$ raises its filtration. Iteration and completeness leave zero. Hence the <primitive elements of a completed rational Iwasawa algebra> satisfy
$$
\boxed{P(\widehat{\mathbb Q_pG})
=\bigoplus_{i=1}^d\mathbb Q_p\log(g_i).}
$$
For the upper-triangular group, ordinary matrix logarithms give
$$
\log\begin{pmatrix}1+a&b\\0&1\end{pmatrix}
=\begin{pmatrix}
\log(1+a)&b\,\dfrac{\log(1+a)}a\\[4pt]
0&0
\end{pmatrix},
$$
with the quotient interpreted as $1$ at $a=0$. For odd $p$, both $a\mapsto\log(1+a)$ and $a\mapsto\log(1+a)/a$ are respectively a bijection $p\mathbb Z_p\to p\mathbb Z_p$ and a unit-valued function. Therefore
$$
\boxed{\log(G)=
\left\{\begin{pmatrix}u&v\\0&0\end{pmatrix}:u,v\in p\mathbb Z_p\right\}.}
$$
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