Sum of Logarithms/Natural Logarithm/Proof 4

Theorem

Let $x, y \in \R$ be strictly positive real numbers.

Then:

$\ln x + \ln y = \map \ln {x y}$

where $\ln$ denotes the natural logarithm.

Proof

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

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

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 $\ln x y$.

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: Exercise $23$