= Solution
Every nonzero <P-adic number> has a unique decomposition $z=p^mu$ with $m\in\mathbb Z$ and $u\in\mathbb Z_p^\times$. The <group homomorphism kernel> of reduction from the <unit group> onto $\mathbb F_p^\times$ is $1+p\mathbb Z_p$, the group of <principal units>. Each nonzero residue class has a unique lift $\omega$ satisfying $\omega^{p-1}=1$, by the simple-root <Hensel lemma> proved in question 3. Uniqueness makes these lifts multiplicative. They form the <cyclic group> $\mu_{p-1}$, and every unit has a unique factorization
$$
u=\omega\,v,\qquad\omega\in\mu_{p-1},\quad v\in1+p\mathbb Z_p.
$$
Thus $\mathbb Q_p^\times=p^{\mathbb Z}\times\mu_{p-1}\times(1+p\mathbb Z_p)$.
For odd $p$, the <p-adic logarithm> and <p-adic exponential function> give inverse group isomorphisms $1+p\mathbb Z_p\leftrightarrow p\mathbb Z_p$. To verify the convergence domain, for $t\in p\mathbb Z_p$ the <p-adic logarithm> terms have valuations $nv_p(t)-v_p(n)\to\infty$, and the exponential terms have valuations $nv_p(t)-v_p(n!)\to\infty$, since $v_p(n!)\le(n-1)/(p-1)$ and $p>2$. In both series every term after the linear one has strictly larger <valuation> than the linear term. Consequently $\log(1+t)\in p\mathbb Z_p$ and $\exp(t)\in1+p\mathbb Z_p$. The formal identities $\log(vw)=\log v+\log w$ and $\exp(\log v)=v$, $\log(\exp t)=t$ hold on these convergent domains, as follows by multiplying the convergent series or passing to their formal identities termwise.
Choose a generator $\zeta$ of $\mu_{p-1}$. An explicit isomorphism is
$$
\boxed{\mathbb Z/(p-1)\mathbb Z\times\mathbb Z_p\times\mathbb Z\longrightarrow\mathbb Q_p^\times,\qquad(a,b,m)\longmapsto\zeta^a\exp(pb)p^m.}
$$
The inverse reads off the <valuation>, the root-of-unity unit factor, and $p^{-1}\log(v)$. Both directions are continuous with the standard product topology, so this is also a <topological group> isomorphism. The odd-prime condition is essential for using all of $1+p\mathbb Z_p$ as the <p-adic logarithm>-isomorphism domain.
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