There is no 'nice' complex logarithm Two problems are meant to establish that no 'nice' logarithm function exists for complex numbers.
The first is
Let $U$ be an open set in $\mathfrak{C}\setminus\{0\}$.
Suppose $h:U \to \mathfrak{C}$ is a continuous function, such that $$e^{h(z)}=z \ \ \ \text{for every} \ z\in U$$
Show that $h$ is holomorphic by computing the defining formula for the derivative. $$lim_{w \to z} \frac{h(w)-h(z)}{w-z}$$
While the second is to prove no such function can exist on $\mathfrak{C}\setminus\{0\}$.
My attempt so far has been to write $h(z)=f(z)+ig(z)$  which then gives $f(z)=\ln|z|$ and $g(z)=\arg(z)$ 
which contradicts the assumption of continuity, proving the second part. This does not prove the first part, which is that if such an $h$ did exist then $h$ is holomorphic

You can differentiate both sides of the relation to obtain that
$$e^{h(z)}\cdot h'(z) = 1 \Longrightarrow h'(z) = \frac{1}{e^{h(z)}} = \frac{1}{z}$$
which I suppose is the formula the question asks for. Notice that $z=0 \notin U$.
