For quaternions, is the natural log the inverse of the exponential function?

Question

That is to ask, is $e^{\ln(q_0)}$ = $\ln(e^{q_0})$?

Answer 1

$e^{\ln(q_0)} = q_0$ always holds for all branches of the logarithm and even for square matrices (because this is essentially the definition of the log).

$\ln(e^{q_0}) = q_0$ is already for complex numbers not always true (because the logarithm is multivalued)

Answer 2

I wrote software to play with quaternions numerically on the command line (http://sourceforge.net/projects/quaternions/).

q_ln 1 2 3 4 | q_exp 1.0000000000000011 2.0000000000000000 3.0000000000000004 4.0000000000000000
This confirms $e^{ln(q_0)}=q_0$. Flip the order: q_exp 1 2 3 4 | q_ln 1.0000000000000000 -0.3335164408915938 -0.5002746613373907 -0.6670328817831875 So $\ln(e^{q_0}) \ne q_0$.