Let $K$ be an algebraically closed field of characteristic zero. I am struggling to solve two exercises in Humphreys Linear algebraic groups: 15.8 and 15.10.
The first asks us to show that the maps
$$\exp(x) =\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ $$\log(x)=\sum_{k=1}^{\infty}(-1)^{(k-1)}(x-\mathbb{1})^k$$
satisfy $\exp(\log(u))=u$ for any unipotent matrix $u$, and $\log(\exp(n))=n$ for any nilpotent matrix $n$ (in this case, both $\exp$ and $\log$ are finite sums). I assume that this is just a matter of working through the algebra to simplify the sums, but I am having trouble with it.
The second asks us to show that for a unipotent closed subgroup $U \subseteq \text{GL}(V)$, and an element $x \in U$, it holds that $\log(x) \in \mathfrak{u}$, the Lie algebra of $U$. I know that $U$ must be connected, and that $\exp \log(ax) \in U$ for all $a \in K$ (but I'm not sure if this helps).