Log-linearization of CES production function

I am trying to recover the Log-linearisation of a CES production function in a paper. Although I am fairly confident with Log-linearisations, I simply do not find the supposed result.

The production function:
$Q_t = Bigg[(1-mu)^{frac{1}{epsilon}}bigg(big(frac{K_t}{alpha}big)^{alpha}big(frac{L_t}{1-alpha}big)^{1-alpha}bigg)^{frac{epsilon-1}{epsilon}}+(mu)^{frac{1}{epsilon}}(M_t)^{frac{epsilon-1}{epsilon}}Bigg]^{frac{epsilon}{epsilon-1}}$

The log-linearization:
$hat{q}_t = alpha(1-S_M) hat{k}_t + (1-alpha)(1-S_M)hat{l}_t + S_M hat{m}_t$
where ($S_M = frac{M}{Q}$)

My approach so far:

$(Qe^{hat{q}_t})^{frac{epsilon-1}{epsilon}} = Bigg[(1-mu)^{frac{1}{epsilon}}bigg(big(frac{Ke^{hat{k}_t}}{alpha}big)^{alpha}big(frac{Le^{hat{l}_t}}{1-alpha}big)^{1-alpha}bigg)^{frac{epsilon-1}{epsilon}}+(mu)^{frac{1}{epsilon}}(Me^{hat{m}_t})^{frac{epsilon-1}{epsilon}}Bigg]$

$(e^{hat{q}_t})^{frac{epsilon-1}{epsilon}} = Bigg[(1-mu)^{frac{1}{epsilon}}bigg(big( frac{K^{alpha}L^{1-alpha}}{Q}big) big(frac{e^{hat{k}_t}}{alpha}big)^{alpha}big(frac{e^{hat{l}_t}}{1-alpha}big)^{1-alpha}bigg)^{frac{epsilon-1}{epsilon}}+(mu)^{frac{1}{epsilon}} (frac{M}{Q})^{frac{epsilon-1}{epsilon}} (e^{hat{m}_t})^{frac{epsilon-1}{epsilon}}Bigg]$

$frac{epsilon_Q-1}{epsilon} (1+hat{q}_t) = (1-mu)^{frac{1}{epsilon}} (1-S_M)^{frac{epsilon-1}{epsilon}}frac{epsilon-1}{epsilon}(hat{k}_t+hat{l}_t)+(mu)^{frac{1}{epsilon}}S_M^{frac{epsilon-1}{epsilon}} frac{epsilon-1}{epsilon}(1+hat{m}_t) $

$(1+hat{q}_t) = (1-mu)^{frac{1}{epsilon}} (1-S_M)^{frac{epsilon-1}{epsilon}}(hat{k}_t+hat{l}_t)+(mu)^{frac{1}{epsilon}}S_M^{frac{epsilon-1}{epsilon}} (1+hat{m}_t) $

What is irritating me the most are the remaining $epsilon$ in the powers of the factor shares. Any hint on how to go further?