Sum of Logarithms/Natural Logarithm/Proof 4

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


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


 * SumOfLogarithmsProof4.png

Consider the diagram above.

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

which is represented by the yellow region above.

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

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:
 * $\left({x, 0}\right), \left({x, 1 / x}\right), \left({x y, \dfrac 1 {x y}}\right), \left({x y, 0}\right)$

which is exactly the green area.

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