Question about Forward Propagation formula

[Kihoon-Shin]

Hello,
I read your blog post about FAST-LIO and I have some questions about formula (28)
$Exp(E)Exp(A(E)^T\Delta E) = d^{-1}\cdot a \cdot Exp(b)Exp(A(b)^T\Delta b)\cdot Exp(c)$

From, $Exp(E) = d^{-1}\cdot a \cdot Exp(b) \cdot Exp(c) $
We can write $Exp(E) \cdot Exp^{-1}(c) = d^{-1} \cdot a \cdot Exp(b)$

Therefore, $Exp(E)Exp(A(E)^T\Delta E) = Exp(E) \cdot Exp^{-1}(c)Exp(A(b)^T\Delta b)\cdot Exp(c)$
$\therefore Exp(A(E)^T\Delta E) = Exp^{-1}(c)Exp(A(b)^T\Delta b)\cdot Exp(c)$

But in you blog post, It is written $Exp(A(E)^T\Delta E) = Exp(-c)Exp(A(b)^T\Delta b)\cdot Exp(c)$
Is it possible to use $Exp^{-1}(c)$ as a $Exp(-c)$ ?