Sum of Logarithms/Natural Logarithm/Proof 4

Proof
Recall the definition of the natural logarithm as the definite integral of the reciprocal function:


 * $\ds \ln x := \int_1^x \frac {\d t} t$


 * SumOfLogarithmsProof4.png

Consider the diagram above.

The value of $\ln x$ is represented by the area between the points:
 * $\tuple {1, 0}, \tuple {1, 1}, \tuple {x, \dfrac 1 x}, \tuple {x, 0}$

which is represented by the yellow region above.

Similarly, the value of $\ln y$ is represented by the area between the points:
 * $\tuple {1, 0}, \tuple {1, 1}, \tuple {y, \dfrac 1 y}, \tuple {y, 0}$

Let the second of these areas be transformed by dividing its height by $x$ and multiplying its length by $x$.

This will preserve its area, while making it into the area between the points:
 * $\tuple {x, 0}, \tuple {x, 1 / x}, \tuple {x y, \dfrac 1 {x y} }, \tuple {x y, 0}$

which is exactly the green area.

The total of the green and yellow areas represents the value of $\ln x y$.