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The bandwidth of a circuit, amplifier etc has been chosen to be defined by the points when the power output falls to half its maximum. Log 0.5 is -0.3 which in the decibel notation is -3dB (-0.3 Bels= -3 decibels). When the output power is at 0.5 maximum, then since the power varies as the square of the voltage, the output voltage is 0.707 of the maximum output voltage.

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