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.
$\log(X + Y) = \log(X) + \log(1 + Y/X)$, so if either $X$ is small compared to $Y$ or vice versa then you can approximate with a Taylor approximation.
Working in other question I found that for positive $x,y$ you could approximate: $$\ln(x+y)\approx \begin{cases} \frac{x^{\frac{1}{\ln(2)}}\ln(x)-y^{\frac{1}{\ln(2)}}\ln(y)}{x^{\frac{1}{\ln(2)}}-y^{\frac{1}{\ln(2)}}}, \quad x\neq y\\ \ln(x)+\ln(2),\quad x=y \end{cases}$$
which is based in this other approximation: $$\ln(1+e^x) \approx \begin{cases} \frac{x}{1-e^{-\frac{x}{\ln(2)}}},\quad x\neq 0\\ \ln(2),\quad x=0\end{cases}$$
For the last one the maximum error is below $0.0092$, and the plot is quite close:
