The Weber-Fechner law is a psychological law states essentially that a person's perception of a change in stimuli (that is, when they notice that a stimuli has changed) is logarithmic with respect to the actual change being produced. Despite its simplistic nature, it has proven applicable to many forms of stimuli, from touch, sight, and even non-physical stimuli, such as market forces and wargame strategies.
The idea that perception of change did not correspond directly with change was first visited by Ernst Heinrich Weber in a series of classic experiments in the late 19th century. He would blindfold a man, place one end of a scale in his hand, and then slowly remove weight from the other end of the scale, thus increasing the perceived weight in the man's hand. He would then ask the blindfolded subject to announce when he could feel the change in weight.
What Weber discovered was that perception was relative in nature. That is, if he doubled the initial weight in the man's hand, he would have to double the weight change to get a reaction from the man. What this meant was the change in perception was equal to the change in the stimulus divided by the stimulus itself times some arbitrary value (determined by the experiment itself.)
Integrating this, and solving for the value at which no change in perception occurs reveals the formula
p = k ln (S / S0)
Where p is perception, k is the arbitrary value, S is the stimulus at the instant, and S0 is the amount of stimulus at which there is no perception at all. In short, this proved that perception is logarithmic to stimulus.
For Weber, this corresponded with other physical stimuli/response systems: in music, the high note of an octave has twice the frequency of the low note, yet the pitch difference is perceived as the same for every octave; in vision, the perceived brightness of stars corresponds logarithmically with the actual intensity; and so on.
Gustav Theodor Fechner first coined the term "Weber's law" to refer to Weber's findings, and suggested that the formula was a mathematical proof of Rene Descartes's mind/body dilemma. Fechner improved upon Weber's findings by impressing upon other psychologists that sensation could be accurately predicted if the stimuli were known.
However, Fechner took this idea one step too far in suggesting that if stimuli could be measured, so too could sensations. He spent the rest of his academic life attempting to do so, but as William James soundly scolded, "Sensations are indivisible units." James offered an alternative theory, that the delay between a change in stimuli and our perception was caused by "friction" in our brain as the result of "shifting gears" between appreciating the state our body is in and the incoming messages indicating changes are occurring.
Despite Fechner's missteps, the law was eventually hyphenated to include his work on the subject, and remains so to this day.
The Weber-Fechner law's initial basis was in physical stimuli, but other scientists found applicability in their own fields as well. Economists generated the concept of marginal utility and the law of diminishing returns in response to the findings of Weber and Fechner; and physicists and chemists both saw a basis in the relativistic nature of perception for further experiments into the mysteries of the atom.
Today, the law has been more or less superseded by Stevens' power law, which states that perception is related to an exponential change in intensity. Stevens' law, like Weber and Fechner before him, does an adequate job of explaining real-world differences in perception between two stimuli, but still lacks a sound theoretical basis on which to proceed.