# Log-linear version of current account equation

I am working on my thesis, doing a two-country model with traded and non-traded goods considering incomplete asset markets and imperfect pass-through. In a sense, I am following Benigno & Thoenissen (2008).

I have been having troubles in log-linearizing my current account equation, which is given by

$$\begin{gather} \frac{S_tB_{F,t}}{P_t(1+i^*_t)} \frac{1}{\Theta\Big(\frac{S_tB_{F,t}}{P_t}\Big)} - \frac{S_tB_{F,t-1}}{P_t} + \frac{P_{H,t}}{P_t}Y_{H,t} + \frac{P_{N,t}}{P_t}Y_{N,t} - Z_t \end{gather}$$

where $$\Theta(\cdot)$$ is the cost function that drives a wedge between the return on foreign currency denominated bonds and $$Z_t = C_t + I_t$$.

I have seen in different papers that it is standard to assume that $$b_t = \frac{S_tB_{F,t}}{P_t}$$ and $$\Theta(b)=1$$, where $$b$$ is the steady state value, and that the steady-state ratio of net foreign assets to GDP, which would be in my case $$\bar{b}/\bar{Z}$$ is assumed to be zero.Let $$\varepsilon = -\Theta '(b)Z$$.

Moreover, I have seen that it is standard to say that $$\hat{b}_t$$ is the deviation of foreign currency denominated bond holdings from their steady state, relative to domestic GDP. Hence, the log-linearized expression of the current account takes the form of

$$\begin{gather} \beta \hat{b}_t = \hat{b}_{t-1} + ... \end{gather}$$

My question is how can I log linearize the left hand side of my current account expression? Because from what I have read, a normalization with the GDP is used, so $$\hat{b}_t$$ is not a typical log-linearization per se.

I would appreciate very much any help.

Alejandro