Car batter was 5V and I used noco boost GB70 to jump start it. But I hear that if the car battery is low, you don't wanna recharge is by running the car as it might damage the alternator. However, we are supposed to use the booster precisely when the battery is low. Then how am I supposed to recharge the batter now? If I need a car battery charger to charge it then what's the point of jump-starting using the booster as the charger would have brought up the charge to a level where the car could be started anyway?
But I hear that if the car battery is low, you don't wanna recharge is by running the car as it might damage the alternator.
Not just the alternator. In fact, you can damage quite a lot of expensive electronics in the car without even damaging the alternator, if running the car with a broken battery, by jump starting it. I suspect those incidents are forcing many car manufacturers to recommend NOT to jump start a modern car. (Hint: even if the manufacturer recommends against jump starting, you can "jump start" the car by rapidly charging its battery from the helper car, disconnect the cables, and then try to start the car with possibly damaged battery, without any danger of breaking anything apart from connecting the cables in incorrect configuration or otherwise creating a short circuit.)
The battery is the buffer for the electrical system in the car. Without it in working order, the alternator can create voltage spikes. Those voltage spikes can and will damage each and every sensitive electronics component in your car. With a new expensive premium car, I can easily see the damage exceeding $10000.
Thus, the recommended action when suspecting a broken battery is to charge the battery, and then try to start the car without any external help by just using the battery. If the battery is broken, it won't start the car at all. No room for expensive damage.
When the battery is so low then the alternator may not even sense sufficient voltage to start charging.
What you should do is to charge the battery from an external charger for 12 to 24 hours and then check the vehicle starts and the charging system works correctly.
However, if the battery was that low for an extended period it may have failed, or it may have an internal fault.
My first step - charge the battery and see.