1. Choose LED current
e.g. I(led) =20mA
2. Choose collector resistor (Rc) value:
Vsupply=LED supply voltage, e.g.: 5V
Vf= LED forward voltage drop, e.g.:2V
Vr=collector resistor voltage drop
Vr=Vsupply-Vf
Rc*I(led)=Vsupply-Vf
Rc=(Vsupply-Vf)/I(led)
Rc=(5V-2V)/0.020=150Ohm
3. Choose base resistor (Rb) value (using 10x base current for saturation) Assuming that the "control voltage" is the same as the supply voltage.
Ibase*Hfe=I(led) * 10
Vsupply/Rb = I(led) * 10
Rb = (Vsupply * Hfe) / ( I(led) * 10 )
Rb = (5V*25)/(0.005*10) = 2.5K