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Mathematica can easily compute exponentials and logarithms of concrete quaternions:

Needs["Quaternions`"]

Exp[Quaternion[1.3, 3.5, -0.7, 0.9]]
Log[Quaternion[1.3, 3.5, -0.7, 0.9]]
Quaternion[-3.14824, -1.79206, 0.358412, -0.460816]

Quaternion[1.36196, 1.17075, -0.23415, 0.301051]

Even for exact numbers:

Exp[Quaternion[1, 3, 7/2, -5]]
Quaternion[E Cos[Sqrt[185]/2], (6 E Sin[Sqrt[185]/2])/Sqrt[185],
           (7 E Sin[Sqrt[185]/2])/Sqrt[185], -2 Sqrt[5/37] E Sin[Sqrt[185]/2]]

But it seems to be unable to expand this for the general quaternion:

Exp[Quaternion[a, b, c, d]]
E^Quaternion[a, b, c, d]

I've tried using Simplify, FunctionExpand, ToQuaternion on this, but still it doesn't give a Quaternion object. But there exists a closed form for such functions.

I can, of course, define such functions myself, like:

exp[q_] = With[{v = q - Re[q]}, Exp[Re[q]] (Cos[Abs[v]] + v/Abs[v] Sin[Abs[v]]) ];
ln[q_] = Log[Abs[q]] + (q - Re[q])/AbsIJK[q] ArcCos[Re[q]/Abs[q]];

But is there a way to get Mathematica itself expand these functions of general quaternions, without me having to redefine them all?

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The reason for the behavior you were seeing can be seen if you look through the definitions within the package. In particular, the extended elementary functions are defined with a format like func[a:Quaternion[__?ScalarQ]] := (* stuff *) where ScalarQ[] is a private package function that checks if the stuff within the Quaternion[] object are real numbers. Since the components of Quaternion[a, b, c, d] are manifestly not real (in the sense of not satisfying ScalarQ[]), the formulae within are not applied to your symbolic quaternion.

But, since this is a package function, you can modify the code yourself so that it can work with quaternions with symbolic components.

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