Consider $\alpha = \log a$ and $\beta = \log b$, $b>a$. Are there formulas for approximating $\gamma = \log (a+b)$? What about $\theta = \log (a-b)$?
If it makes it easier, assume that $|\alpha| \gg 300$ so the obvious solution $\gamma = \log (10^\alpha + 10^\beta)$ is infeasible using standard double-precision floating point arithmetic (you can't easily store $a$ and $b$ in memory to begin with!).