I am learning about the log map on $SE(3)$ and I want to check my understanding of properties for use in solving an equation. Are the following true, for A, B, C as elements of $SE(3)$?
$$ \log(ABC) = \log A + \log B + \log C $$
$$ \log(e^{A}) = A $$
$$ \log(e^{At}) = At $$ for $t$ as a real number.
In particular I want to solve the equation
$$ e^{A_1t_1}e^{A_2t_2}e^{A_3t_3} = g $$ for scalars $t_1, t_2, t_3$ with $A_1, A_2, A_3, g$ in $SE(3)$. My plan is to use the log mapping to get an easier linear matrix equation to solve.
Thanks for any help!