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What is the relationship between the intensity of the stimulus and the intensity of the sensation as represented by Fechner's equation S K log R?

Sensation increases as the logrithm of stimulus intensity or "S = k log R".

What is the relationship between the intensity of the stimulus and the intensity of the sensation?

Intensity of a sensation is directly proportional to the intensity of the physical stimulus raised to a constant power.

How does the difference threshold for a stimulus relate to the intensity of the stimulus?

The Difference Threshold (or "Just Noticeable Difference") is the minimum amount by which stimulus intensity must be changed in order to produce a noticeable variation in sensory experience.

Which theory proposes that as physical stimulus intensity R grows larger the sensation intensity's increases logarithmically?

Fechner's law states that the subjective sensation is proportional to the logarithm of the stimulus intensity.