Whether conversing in a noisy room, savoring an exotic dish or looking out for fast movement in a dark alley, the ability to sense the external world is critical to our happiness and survival. Psychophysical experiments suggest that perception is often logarithmically related to stimulus intensity. We formulate a Bayesian framework where psychophysical scales arise as optimizations under cognitive limitation and expected relative error (ERE) fidelity. We consider models where latency and storage are the limiting resources and, using analyses related to high-resolution quantization theory, find that the logarithmic relationship appears naturally. We demonstrate the validity of our model for natural sound data sets and discuss implications of this work in psychophysics and source coding.