Probability and log probability in hidden Markov models

I have a set of Observation Symbol Sequences which I have to test against a set of Trained HMM classifiers. I seem to understand the advantages of using Log Probability over regular probabilities.

In the testing phase of a HMM classifier, I don't seem to get the motivation behind multiplying probabilities or adding log probabilities in determining the class of the observed observation test sequence.

Why do we have to multiply or add probabilities? Can't we just determine probability or log probability of a Single Observation Symbol Sequence using Forward algorithm?

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