Please find below an example from the book Visualizing quaternion of Andrew Hanson:
Let's say I have a complex number to a certain power:
$$(e^{iθ})^t=e^{itθ}$$
let's consider the derivative rule $ \frac{d e^t}{dt}$, then: $$ \frac{d \ e^{itθ}}{dt}=iθe^{itθ}$$
And the derivative rule $ \ln{x}=\frac{dx}{x}$, then: $$ \frac{d \ ln{\ e^{itθ}}}{dt}=iθ$$
This examples is in the paragraph about expoential map and tangent space. As far as I understand it, this should be the proof of the logarithm being a way to map a point on the circle to an infinite amount of points in the real axis and that can be extended to quaternion to linearize them locally and being able to invert the mapping with the exponential. He doesn't say so explicitly, but this is the conclusion I get. First question, is my conclusion right?
Second, how can I extend this proof to quaternions? I mean, given a quaternion in exponential form $e^{n\frac{\theta}{2}}$, how can I derive its logarithm to prove that the derivative is constant?