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Was the battery light on while the engine is running?
There are a few things that can affect the alternators performance, The alternator itself obviously, The alternator harness (the cable that runs from the alternator to the fuse panel ontop of the battery, and the alternator pulley.
Assuming the alternator and pulley are the original ones in the car i would have your mechanic do a voltage drop across the alternator harness. If he is not familar with such a technique i can assist you on how to measure this.
You will want to locate the alternator and the large cable running to it (there will also be a connector with two wires going into it) remove the little cap that is covering the nut holding the harness onto the alternator. you will want to place one digital voltmeter lead on this nut. The other lead of the Voltmeter will want to go on the most left cable on the fuse panel ontop of the battery, or the positive terminal on the battery, the reading should not be more than .3V if its more i would suggest replacing the Alternator harness as well as the alternator, and retain the OEM pulley.
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Alright, well not necessarily will the engines computer cause any sort of issue like this.
With your explaination above it sounds like you have a bad battery as well as a faulty alternator.
The only reason for a concern with the engines computers causing larger issue's, would be if the car has a electrical draw. Although, the issue you described it sounds more like a charging system issue, as long as your battery isnt dead in a week you should be ok.
To be certain and put your mind to ease, after the charging system issue is resolved you can take the car to the dealer or any good diagnostic shop and have a parasitic draw test performed to see if any computers are draining the battery.
Goodluck with the repairs, and make sure firestone uses the OEM pulley with the alternator they install.
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When and If you go in to the dealer after the alternator replacement, Just tell them that you have just replaced the alternator and battery, and would like any faults cleared, and a parasitic amperage draw test (the tecnicians will know what you are refering to, the service advisor might not) performed, ask them for a printout of the readings of the draw test, just for proof they have actually performed the test.
You most likely would be looking at about 1hr of labor for this, and depending on the dealer you take your car to thier labor rate should be in the ballpark of $110.
Goodluck with the repair and if you have anymore concerns feel free to ask!
Feedback is appreciated!