Logarithmic Property of the Quadrature of the Hyperbola
For , the natural logarithm is the signed area under the curve between 1 and . Its essential property, that , arises from area-preserving symmetries of the curve.
A transformation of the form is area-preserving: any region is transformed to a region with the same area, as may be confirmed by applying it to a rectangle with sides parallel to the axes. It is also a symmetry of : any point on the curve is mapped to another point on the curve. Moving the "slide" controller shows the effect of a continuous family of such transformations between the identity and acting on the region under the curve over the interval . With the checkbox you can see the effect on a rectangle placed under the curve.
If denotes the area under the curve over an interval , the area-preserving and symmetry properties of tell us that . This is seen by varying "slide" through its range. Thus