I have a doubt on how to log linearize an expression for the real exchange rate. I have read in several papers that they express it using substitutions. I haven't been able to find out what substitutions they use, so that is why I come here to ask for help.
For example, suppose that the law of one price holds such that:
\begin{gather} P_{H,t}=S_tP^*_{H,t} \quad \quad \text{and} \quad \quad P^*_{F,t}=\frac{P_{F,t}}{S_t} \end{gather}
Let $Q_t=\frac{S_tP^*_t}{P_t}$ be the real exchange rate and $T_t=\frac{S_tP^*_{F,t}}{P_{H,t}}$. Moreover, let the CPI be
\begin{gather} P_t=[aP^{1-\theta}_{H,t}+(1-a)P^{1-\theta}P_{F,t}]^{\frac{1}{1-\theta}} \end{gather}
I know that the log-linear version of $Q_t$ is given by
$$\hat{Q_t}=(2a-1)\hat{T}_t$$
Can someone please tell me how to obtain this latter expression?
Thank you very much.