How to compose motors, given as exponentials? (Lie algebra or screw theory?)

Dear community!
I’m having problem to figure out how to express log of motor after composition of 2 exponentials:

exp(C) = exp(A)*exp(B)
C = ?
(A, B are given bivectors, and C is required one)

It looks like I have to go to the screw theory, and decompose A into (-φ La/2 + δLa I/2), and B as (-φ Lb/2 + δLb I/2)
But after that have no clue how to proceed.
Can you please suggest either a solution or a path what to do?
If my assumption of screw theory need is correct, then it would make sense to get obtain screw representation of C, and I’ll be happy.
I’m more of a practitioner, not so good at theory.

PS

Of course I may go and calculate log(exp(A)*exp(B)) by some clever formula. The reason I would like to have a direct expression is that I want to use differentiation of the result.
So, my problem might sound even like that:
exp(A1) = exp(A0)exp(Bt)

Actually, I have one more question.
How many degrees of freedom in this motor representation exp(-φ L/2 + δL I/2)? Is L is a normal line with 3DoF or general one with 6DoF?