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I know my LED has a voltage drop of 1.35V and I want to run it at 100mA and that I'll be supplying it with 5V from the Arduino. What I don't understand is the math for how to figure out the exact voltage drop of the transistor between the collector and emitter. And I'm also trying to figure out the math used to calculate the required milliamps that have to flow through the base of the transistor in order to fully turn it on (but not waste extra electricity).

I know that there is quite a lot of lee way in which resistors to use and the circuit will still work, but I'm hoping to figure out the math so that I can get as close as possible to using the exactly perfect resistor values.