The logarithm of a general quaternion is defined as $$log(q) =\left (\left|q\right|, \frac{\mathbf{v}}{\left|\mathbf{v}\right|}cos^{-1}\left(\frac{r}{\left|q\right|}\right)\right),$$
in $(r,\mathbf{v})$ notation, where $r \in \mathbb{R}$ is the real part, and $\mathbf{v} \in \mathbb{R}^3$ is the vector part of the quaternion. The exponent is defined accordingly, so that supposedly $exp(log(q))=q$, no matter what branch we choose for $log(q)$.
However, for real quaternion ($\mathbf{v}=\mathbf{0}$), there is a definition problem. Even if we suppose that the undefined $\frac{\mathbf{v}}{\left|\mathbf{v}\right|}$ is just $\mathbf{0}$, we get $log(q)=(log\left|q\right|,\mathbf{0})$, which definitely does not reproduce $q$ with the exponent. It seems we lose the real part quantification this way. For instance,
$$log(-1,0,0,0)=(0,0,0,0),\ exp(0,0,0,0)=(1,0,0,0).$$
Is there an alternative or more general definition that fixes this?