Derivation of quaternion logarithm

Question

I'm trying to understand how the inverse of the quaternion exponential was derived. Given the definition of the quaternion exponential, $$e^Q=e^{a+bi+cj+dk}=e^{a+v} = x+yi+zj+wk = e^a\frac{v}{|v|}\sin(|v|),$$ we can expand it to see $$x+yi+zj+wk = e^a\cos(|v|)+\frac{e^a\sin(|v|)b}{|v|}i + \frac{e^a\sin(|v|)c}{|v|}j+\frac{e^a\sin(|v|)d}{|v|}k,$$ showing $x=e^a\cos(|v|)$, $y=\frac{e^a\sin(|v|)b}{|v|}$, and so on. Given only that information, it seems impossible to recover $a,b,c,d$ since they are all dependent on $|v|$, which appears inextricable as it is tied up in three terms as $\frac{sin|v|}{|v|}$ which has no simple inverse. The quaternion logarithm must have been found a different way. I am aware that a perfectly good logarithm for quaternions exists, but I want to know how the exponential function was inverted to find it. I have looked online and only found "the quaternion logarithm is defined as ..." without ever finding a proof as to why this is the definition.

Answer 1

Just like the complex exponential the map $Q \mapsto e^Q$ isn't really invertible. You have to restrict the kinds of $Q$ you consider to get a true inverse.

Let $P = x + yi + zj + wk$. Usually we just normalize $u = (P-\bar P)/2 = yi + zj + wk$ to get $v' = u/|u|$, calculate $\theta$ by solving the equations $|P|\cos\theta = x$ and $|P|\sin\theta = |u|$, and set $a' = \ln|P|$. Then it follows easily that $e^Q = e^{a' + \theta v'}$: $$ e^{a' + \theta v'} = |P|(\cos\theta + v'\sin\theta) = x + v'|u| = x + u = P. $$ If $|u| = 0$ and $x > 0$ then we take the logarithm to be $\ln P$. If $|u| = 0$ but $x < 0$ then we can take the logarithm to be $\ln|P| + \pi v'$ for abitrary unit $v'$.