Rational Points on Graph of Logarithm Function

Theorem

Consider the graph of the logarithm function in the real Cartesian plane $\R^2$:

$f := \set {\tuple {x, y} \in \R^2: y = \ln x}$

The only rational point of $f$ is $\tuple {1, 0}$.

Proof

Consider the graph of the exponential function in the real Cartesian plane $\R^2$:

$g := \set {\tuple {x, y} \in \R^2: y = e^x}$

From Rational Points on Graph of Exponential Function, the only rational point of $g$ is $\tuple {0, 1}$.

By definition of the exponential function, $f$ and $g$ are inverses.

Thus:

$\tuple {x, y} \in g \iff \tuple {y, x} \in f$

Thus for $\tuple {x, y} \in g$ to be rational, $\tuple {y, x} = \tuple {0, 1}$.

Hence the result.

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.17$: More About Irrational Numbers. $\pi$ is Irrational