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I'm reading 3D math primer for graphics and game development by Fletcher Dunn and Ian Parberry. On page 170, the logarithm of quaternion is defined as

\begin{align} \log \mathbf q &= \log \left( \begin{bmatrix} \cos \alpha & \mathbf n \sin \alpha \end{bmatrix} \right) \\\\ &\equiv \begin{bmatrix} 0 & \alpha \mathbf n \end{bmatrix} \end{align}

I don't see how $\log \left( \begin{bmatrix} \cos \alpha & \mathbf n \sin \alpha \end{bmatrix} \right)$ is equal to $\begin{bmatrix} 0 & \alpha \mathbf n \end{bmatrix}$. Can anyone help me out?

Thanks.

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    I don't see how you can have a log of a quat, since quat addition is commutative but multiplication not. The standard rule for logarithms, log(ab)=log(a)+log(b) would not apply except over a subset where log(ab)=log(ba). Sorry, Lie algebra is above my paygrade. There are other unreal objects that have perfectly valid logarithms, like elements of a finite field with a single generator of the multiplicative group. I have in mind the Hebrew alphabet and the practice of gematria, as it maps onto the integers 1 to 22, and arithmetic is done modulo 23.2017-07-21

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