I have the following situation:
Now imagine that I do not know the numbers on either side of the $+$-sign, but I only know the corresponding logarithmic values.
So the left side is $\approx -3.79$ and the right side is $ \approx -4.42$ (natural logarithm)
Based on these numbers, I would like to find out the natural logarithm that corresponds to $\log(0.15*0.5*0.3 + 0.1*0.4*0.3)$
How do I do this? I am completely lost. I am trying to implement the Forward algorithm and I would rather deal with summing values rather than multiplying small probabilities to deal with underflow.
If I understand your question correctly:
Let $x = 0.15\times 0.5\times 0.3$ and $y = 0.1\times 0.4\times 0.3$.
$$\begin{align}\ln x \simeq -3.79&\implies x \simeq \exp(-3.79) \\ \ln y \simeq -4.42 &\implies y \simeq \exp(-4.42)\end{align}$$
Therefore,
$$\ln(x + y) \simeq \ln\left(\exp(-3.79) + \exp(-4.42)\right)$$