Gustav Theodor Fechner, in his 1860 masterwork Elemente der Psychophysik, derived a mathematical relationship between physical stimulus intensity and subjective sensation by integrating Weber's Law. If each JND represents an equal subjective step, and if the JND is proportional to stimulus intensity (Weber's Law), then sensation S grows as the logarithm of intensity I.
Derivation
Integrating: S = c · ln(I) + C
Setting S = 0 at threshold I₀:
S = k · ln(I / I₀)
The logarithmic function means that sensation increases rapidly at low intensities but progressively more slowly at high intensities. Doubling the intensity produces the same incremental change in sensation regardless of the starting level. This compressive transformation allows the sensory systems to operate across an enormous dynamic range — the human auditory system, for example, spans roughly 12 orders of magnitude in sound pressure.
Legacy and Critique
Fechner's Law was the first quantitative psychophysical law and established psychophysics as a mathematical science. However, S.S. Stevens argued in the 1950s that Fechner's assumption of equal JND steps was unfounded, proposing instead that direct magnitude estimation reveals a power law relationship. The debate between logarithmic and power law formulations continues, with modern work suggesting that both may be special cases of more general models, and that the appropriate law depends on the experimental method used to measure sensation.