I know there is no formula to separate the log of a sum, e.g. $\log(X+Y)$ into two parts, but are there any approximation rules that can allow me to achieve this objective?
$$E_t(1+r_{t+1}^K)=E_t\left[
\dfrac{\frac1{X_{t+1}}\alpha A_0\frac{Y_{t+1}}{K_{t+1}}+Q_{t+1}(1-\delta)}{Q_t}
\right]$$
Suppose we ignore the expectations operator for the moment.
Best Answer
$\log(X + Y) = \log(X) + \log(1 + Y/X)$, so if either $X$ is small compared to $Y$ or vice versa then you can approximate with a Taylor approximation.