Working in $\mathbb{Q}_p$, with $p\ge 3$ prime. I want to prove that the closure of the cyclic group $(\langle 1+p \rangle,\cdot)$ in the $p$-adic topology is $(1+p\mathbb{Z}_p,\cdot)$, where $\mathbb{Z}_p$ is the ring of integers.
I've already proved that both the structures are groups and that the first is a subset of the second one, so I need to prove the density of one inside the other. I tried fiddling with the explicit power series expression of the elements of $1+\mathbb{Z}_p$ but that didn't lead me anywhere. Any hint?
Hint: Using the logarithm map, for $p\ge3$, $$(1 + p\mathbb Z_p, \cdot) \cong (p\mathbb Z_p, +)$$
Can you show that the image of $(\langle 1+p\rangle, \cdot)$ under this map is $(m\mathbb Z,+)$ for some $m\in\mathbb Z_p$ with $|m|=|p|$ where $m=\log(1+p)$?