Let $A$ be a unital $C^*$ algebra, $u$ an unitary element such that $\sigma(u)\neq\mathbb{T}$ (the spectrum). Consider the usual complex $\log$ function (with the usual branch). Under these circunstances, $\log \in C(\sigma(u))$, so that $\log(u)$ "makes sense". Now let $a:=\frac{\log(u)}{i}$, and prove that it is self adjoint.
To do this it suffices to check that $[\log(u)]^*=-\log(u)$, but I am little confused on how to compute $[\log(u)]^*$. Can someone help me explain why is this true? Thanks!