Is DL difficult under the group of Unimodular matrices?
You will need to solve discrete log(s) over the field itself at some point to solve the matrix equation. Now, discrete logs over $F_{2^n}$, as opposed to $F_p$ ($p$ prime) are much easier as shown in the work by Joux, but no full proof yet. 

However see, e.g.,here where the abstract states 
"A new index calculus algorithm with complexity L(1/4+o(1)) in very small characteristic" is presented.

The L-notation is clarified in wikipedia, but the notation above essentially means that the complexity is essentially $\exp{[ c_n (\log n)^{1/4} (\log\log n)^{3/4}}]$ so closer to polynomial than exponential and the coefficient $c_n$ goes to zero as $n$ goes to infinity.