Let $A$ be a $C^*$-algebra and $a \in A_+$ be a positive element. I want to show that $a$ has a positive logarithm if $a$ is invertible.
I just see that the usual $\log$ function is continuous on the spectrum of $a$ since $0 \notin \sigma(a)$ but $\log(a)$ needs not to be positive as this would by functional calculus be true if $\sigma(a) \subset [1,\infty)$, which I do not see.
If $a=e^b$ for some positive $b$, then $\sigma(a) = \{e^t : t \in \sigma(b)\} \subset [1,\infty)$ so $a$ must necessarily have its spectrum in $[1,\infty)$ I guess.
As written, the exercise is definitely wrong. Take the C$^*$-algebra $\mathcal A=\mathbb C$, and take $a=1/2$. Then the question is Conway's book claims that there exists a positive element $b$ in $\mathbb C$ such that $e^b=1/2$. Of course, such $b$ does not exist.
Most likely the word "positive" in the question is a typo. In that situation, you can use the continuous functional calculus to obtain the logarithm.