Scaling procedures to address false 0's after multiplying probabilities
A simplee trick (explained here) that let you use the $\log$ approach even when you must sum probabilities.

Problem: $\log(\exp(a) + \exp(b))$ can lead to an underflow, to avoid it you can use this formula:

$\log(x + y) = \log(x) + \log(1.0 + \exp( \log(y) - \log(x) ) )$

Or use another approach:

$\log(\exp(a) + \exp(b)) = \log( \exp(a - C) + \exp(b - C)) + C$

Setting $C = \max(a,b)$

For example: $\log(e^{-120}+e^{-121}) = \log(e^{-120}(e^0 + e^{-1}))= \log(e^0+e^{-1})-120$