PIE isn't particularly relevant, because logic is a borrowed word, not a word that has been transmitted to English by inheritance from PIE.
The Oxford English Dictionary's earliest citation for the noun logic is from 1362 (with the spelling "logyk").
It gives the etymology as follows:
Etymology: < French logique (13th cent.), < medieval Latin logica, < Greek λογική (first found in Cicero; elliptical for ἡ λογικὴ τέχνη, rendered in medieval Latin by ars logica) [...]
The consonant sound used for the letter "g"
As you mention, the pronunciation of the letter "g" in Classical Latin before "i" is reconstructed as a velar plosive [g], so we would expect logica to be pronounced in Classical Latin as something like [ˈlogika] (or possibly, with a more narrow transcription of the vowels, [ˈlɔgɪka]).
But the pronunciation of Latin changed over time: note that the OED entry talks about "medieval Latin" and not Classical Latin. The "palatalization" of velar consonants is a common feature of many traditions for pronouncing Latin.
French "logique" appears to be a learned adaptation of Latin logica (in inherited French vocabulary, Latin intervocalic singleton plosives were lenited).
Wikipedia indicates that the de-affrication of [dʒ] in French is thought to have occurred sometime in the second half of the 13th century, so the loan into English may have occurred before this French sound change, or so soon after it that English speakers still were familiar with the use of the sound [dʒ] as a feature of French pronunciation.
Furthermore, the word has been in English for many centuries, and established English words tend to develop more assimilated pronunciations. It seems possible to me that the word might once have been pronounced with [ʒ] by some speakers under the influence of the de-affricated pronunciation that came to be used in French, but if so, that pronunciation must have been lost over time.
The vowel sound used for the letter "o"
It seems fairly clear that the use of an unrounded low vowel [ɑ] for the "short o" sound in (most accents of) American English is a fairly recent development: the "short o" vowel was originally rounded, as it still is for many British English speakers (which is the reason that British English transcriptions typically use the symbol ɒ, although some phoneticians such as Geoff Lindsey have argued that for certain British accents, the symbol ɔ would be more phonetically accurate, and more in line with cross-language usage of IPA symbols).
The Middle English pronunciation of the "short o" vowel is reconstructed as a rounded back vowel closer than [a]. (As far as I know, there isn't total certainty about whether it was closer to IPA cardinal [o] or IPA cardinal [ɔ]: Middle English did not have multiple "short o" sounds of varying heights, although there was a height contrast for "long o" in Middle English. That said, the standard transcription seems to be o, and since this would also suffice as a broad transcription for [ɔ], I will use [o] in the next paragraph to represent the Middle English "short o" sound.)
French and Latin "o" sounds usually correspond to Middle English "short o" (ME [o], Modern AmEng [ɑ]) or "open long o" [ME [ɔː], Modern AmEng [oʊ]).
The use of the "short" value rather than the "long" value in this word is a bit tricky to explain, and I don't know enough to give a confident summary of its history. From a synchronic perspective, the use of a "short" vowel here is regular because this is an "-ic" word, and these words show a strong tendency to have penultimate stress and a weaker tendency (apparent in words like manic, lyric and static—contrast mania, lyre and stasis—but violated in many words like basic and psychic) for a single vowel letter in the stressed syllable to be "short" (unless it is "u", in which case the rule that "u" is always "long" in orthographically open syllables takes precedence). The historical origin of this rule about the modern English pronunciation of -ic words may be influence from the so-called "traditional English pronunciation" of Latin forms like logicus and logica: according to this system, the vowel in the first, stressed syllable of these words is pronounced "short" because it is followed by two other syllables.
