A quaternion encodes uniform scaling + rotation. The logarithm of a quaternion is its axis-angle-nepers form, and vice versa.

    quat = sqrt( exp( nepers + radians * <axis> ) )

Heard the word 'nepers' after many decades. Are you by any chance an Electrical major ?

Thanks for your comment. To be fair, I had not done due diligence before asking. There's a Wikipedia pages on quaternion calculus.

Complex analysis (calculus on functions from 2D rotations to 2D rotations) is beautiful -- Once differentiability guarantees infinite differentiability. Wondering what would the analogue of that be for quaternions