When you connect the emitter through a resistor, the collector to the voltage source and apply enough voltage at the base for conduction, the voltage across the emitter resistor will be roughly equal to the base voltage minus the base emitter junction voltage (0.7v typical for silicon transistors).

The voltage across the emitter resistor is independent of resistor value, so it can be used to power high current drawing loads from a small input current, since the current through the emitter resistor and anything connected to it comes largely from the collector current instead of base current.

This also isolates parts of a circuit from loading or otherwise interfere with it's function. As such, they are also called voltage buffers and impedance buffers.

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