MatrixExp[a(({{0, 0}, {1, 0}}))].MatrixExp[b(({{0, 1}, {0, 0}}))] // MatrixForm

MatrixExp[a(({{0, 0}, {1, 0}}))].MatrixExp[b(({{0, 1}, {0, 0}}))] //MatrixLog//Simplify// MatrixForm

eqn=MatrixExp[a(({{0, 0}, {1, 0}}))].MatrixExp[b(({{0, 1}, {0, 0}}))]==MatrixExp[β(({{0, 1}, {0, 0}}))].MatrixExp[α(({{0, 0}, {1, 0}}))].MatrixExp[γ(({{1, 0}, {0, -1}}))]

Solve[Thread[Flatten/@eqn], {α,β,γ}]

Thread[Flatten/@eqn] /. {α->a (1+a b),β->(b/(1+a b)),γ->Log[(1/(1+a b))]}

ρh=(({{0, 0, -1}, {0, 0, 0}, {0, 0, 0}}));ρe=(({{0, 0, 0}, {0, 0, 1}, {0, 0, 0}})); ρl=(({{0, 0, 0}, {0, 1, 0}, {0, 0, 0}})); ρf=(({{0, 1, 0}, {0, 0, 0}, {0, 0, 0}})); ρ0=(({{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}));

eqn2=MatrixExp[a ρf].MatrixExp[b ρe]==MatrixExp[β ρe].MatrixExp[α ρf].MatrixExp[γ ρh]

Solve[Thread[Flatten/@eqn2], {α,β,γ}]