I have an issue with an exercise that is concerned with the representation of continuous paths on the unit circle.
I started by considering the complex logarithm for $z=e^{i \varphi}, \ \varphi \in (-\pi,\pi]$. Then the complex logarithm can be defined as
$$ \log(z):=\log(|z|)+i \varphi. $$
So any function $\varphi$ that has the required property must also satisfy
$$ \varphi(t)=-i \log(\gamma(t)). \tag{1} $$
I am however a bit confused about how to define $\varphi$ such that $\varphi([0,1] \subset \mathbb{R}$. Of course I could take (1) as the definition. Then $\gamma(t)$ has a polar representation such that
$$ \log(\gamma(t))= \log(1)+i \phi=i \phi, \ \phi \in \mathbb{R}. $$
This would give me the desired result. However this seems to be a bit too simplistic. Am I on the right track or is this supposed to be solved in a different way?