Log linearization

Question

Consider some time series data $X=\{x_t:t\in[0,\infty)\}$. Define a steady state by \begin{align} x\in X:x_{t+1} - x_t = 0 \end{align}

and the log deviation from steady state with \begin{align} \hat x_t:=\ln x_t - \ln_x. \end{align}

Now we can express $x_t$ as \begin{align} x_t=x\exp(\hat x_t). \end{align}

Consider the the following equation \begin{align} a_t = b_t c_t^\alpha + D\mathbb{E}_t\left[\left(\frac{e_{t+1}}{e_t}\right)\left(\frac{f_t}{g_t}\right)^\beta a_{t+1}\right] \end{align} where $D$ is some constant, which is ought to be log linearized for every time dependent variable around steady state.

First Attempt

Express every variable by its new definition,i.e. \begin{align} a\exp(\hat a_t) = b c^\alpha \exp(\hat b_t + \alpha\hat c_t)+ D\left(\frac{f}{g}\right)^\beta a\mathbb{E}_t\left[\exp(\hat e_{t+1} - \hat e_t + \beta(\hat f_t - \hat g_t) + \hat a_{t+1})\right] \end{align}

which should simplify to \begin{align} \exp(\hat a_t) = \exp(\hat b_t + \alpha\hat c_t)+\mathbb{E}_t\left[\exp(\hat e_{t+1} - \hat e_t + \beta(\hat f_t - \hat g_t) + \hat a_{t+1})\right] \end{align}

• Is this correct?
• What can be done to further improve/simplify the equation at hand?