As discussed in Exponential and Logarithm in SO(3), SE(3), Quaternion Space, the formulas are as follows. $$ \log(R) = [\hat{\omega}]\theta $$ $$ e^{[\hat{\omega}]\theta} = R $$
Also as discussed in Exponential and Logarithm in SO(3), SE(3), Quaternion Space, the formulas are as follows.
Given a unit quaternion $ q = \begin{bmatrix} \text{cos}\frac{\theta}{2} & \hat{\omega} \, \text{sin}\frac{\theta}{2} \end{bmatrix} \in \mathbb{H}_1 $, its logarithm is defined as $$ \text{log}(q) \triangleq \begin{bmatrix} 0 & \frac{\theta}{2} \hat{\omega} \end{bmatrix} \in \mathbb{H} $$
For the special case $ q = \begin{bmatrix} 1 & \mathbf{0} \end{bmatrix} \in \mathbb{H}_1$, its logarithm is defined as $$ \text{log}(\begin{bmatrix} 1 & \mathbf{0} \end{bmatrix}) \triangleq \begin{bmatrix} 0 &\mathbf{0} \end{bmatrix} \in \mathbb{H} $$
Logarithm and exponential are inverse of each other. Given the definition of unit quaternion logarithm, the definition of pure quaternion exponential follows naturally.
Given a pure quaternion $\mathcal{Q} = \begin{bmatrix} 0 & \frac{\theta}{2} \hat{\omega} \end{bmatrix} \in \mathbb{H}$, its exponential is defined as $$ \text{exp}(\mathcal{Q}) \triangleq \begin{bmatrix} \text{cos}\frac{\theta}{2} & \hat{\omega} \, \text{sin}\frac{\theta}{2} \end{bmatrix} \in \mathbb{H}_1 $$
For the special case $\mathcal{Q}=\begin{bmatrix} 0 & \mathbf{0} \end{bmatrix} \in \mathbb{H}$, its exponential is defined as $$ \text{exp}(\begin{bmatrix} 0 & \mathbf{0} \end{bmatrix}) \triangleq \begin{bmatrix} 1 & \mathbf{0} \end{bmatrix} \in \mathbb{H}_1 $$