evaluating log(a/b) == log(a) - log(b)

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You have to keep in mind that when checking for an identity, Sage returns True if it has been capable to prove the identity and False if either (i) Sage has been capable to disprove it or (ii) Sage has not been capable to prove or disprove it. So, a False output means either "false" or "unknown". For instance, on your example, we have

sage: bool(log(a / b) == log(a) - log(b))
False
sage: bool(log(a / b) != log(a) - log(b))
False

These two results seem contradictory, but here the two False actually mean "unknown". This state of affairs is due to bool implementing a bivalent logic, not a ternary one (which would have permit Unknown as a possible output).

As @Emmanuel_Charpentier explains in his answer, you have to help Sage by invoking log_expand:

sage: stmt = log(a / b) == log(a) - log(b)
sage: bool(stmt.log_expand())
True

WorksForMe(TM) on Sage 9.8.beta1:

sage: var("a, b")
(a, b)
sage: log(a/b).log_expand()
log(a) - log(b)
sage: log(a/b).log_expand().log_simplify()
log(a/b)

HTH,