True or false logarithmic branches

Question

Say whether the following are true or false. Give a short proof.

1) $log(-z)+i{\pi}$ is a branch of the logarithmic function whose branch cut is the non-negative real axis

2)If $g(z)$ is a branch of the logarithmic function with domain D and $h(z)$ is a branch analytic in D, then there is an integer $m$ with $h(z)=g(z)+2m{\pi}i$

Thanks

Answer 1

Hint for part $a$: if $z=1$, what happens?

Hint for part $b$: if $g$ and $h$ share the same domain, what do you know about the branch cuts in relation to each other? How does this relate to the integer multiples of $2\pi i$?