The logarithm of a quaternion $q$ that has a real part $a$ and imaginary parts $v$ is defined as
$$ \ln q = \ln a + \hat{v} \arccos \frac{a}{\left\lvert q \right\rvert} $$
The exponentional of a quaternion $q$ that has a real part $a$ and imaginary parts $v$ is defined as
$$ \exp q = e^a \left(\cos \left\lvert v\right\rvert+ \hat{v} \sin \left\lvert v \right\rvert\right) $$
Is it true that for all purely imaginary quaternions that:
$$ \ln (\exp u * \exp v) = u + v $$ $$ c u = \ln (\exp u)^c $$
I have a math program I wrote which seems to be inaccurate and want to know if the bug is in my program or in my idea of quarternions.
