In their natural environment, cells need to extract useful information from complex temporal signals that vary over a wide range of intensities and time scales. Here, we study how such signals are processed by Escherichia coli during chemotaxis by developing a general theoretical model based on receptor adaptation and receptor-receptor cooperativity. Measured responses to various monotonic, oscillatory, and impulsive stimuli are all explained consistently by the underlying adaptation kinetics within this model. For exponential ramp signals, an analytical solution is discovered that reveals a remarkable connection between the dependence of kinase activity on the exponential ramp rate and the receptor methylation rate function. For exponentiated sine-wave signals, spectral analysis shows that the chemotaxis pathway acts as a lowpass filter for the derivative of the signal with the cutoff frequency determined by an intrinsic adaptation time scale. For large step stimuli, we find that the recovery time is determined by the constant maximum methylation rate, which provides a natural explanation for the observed recovery time additivity. Our model provides a quantitative system-level description of the chemotaxis signaling pathway and can be used to predict E. coli chemotaxis responses to arbitrary temporal signals. This model of the receptor system reveals the molecular origin of Weber's law in bacterial chemotaxis. We further identify additional constraints required to account for the related observation that the output of this pathway is constant under exponential ramp stimuli, a feature that we call "logarithmic tracking."