Suppose $ A$ is a C*-algebra. Show that an invertible hermitian element of $A$ has a logarithm. ($a$ has a logarithm if there is an element $b\in A$ such that $e^b=a$)
If $a\in A_+$ then it's easy for me to show it, but if $a$ is self-adjoint, then $\log$ function is not continuous on $\sigma(a)$. How can come up by this? Please help me. Thanks.