The use of quantum-mechanically entangled light to illuminate objects can provide substantial enhancements over unentangled light for detecting and imaging those objects in the presence of high levels of noise and loss. Each signal sent out is entangled with an ancilla, which is retained. Detection takes place via an entangling measurement on the returning signal together with the ancilla. This paper shows that for photodetection, quantum illumination with m bits of entanglement can in principle increase the effective signal-to-noise ratio by a factor of 2m, an exponential improvement over unentangled illumination. The enhancement persists even when noise and loss are so great that no entanglement survives at the detector.