User:Keith.U/Sandbox/Proof 1

Theorem
Let $x \in \R$ be a real number.

Let $\ln x$ be the natural logarithm of $x$.

Then $\ln x$ is well-defined.

Proof
This proof assumes the definition of $\ln$ by a definite integral.

Fix $x \in \R_{>0}$.

Let $x \geq 1$.

From Identity Mapping is Continuous and  Combination Theorem for Continuous Functions: Quotient Rule, $\dfrac{1}{x}$ is  continuous on $\left[{1, \,.\,.\, x}\right]$.

From Continuous Function is Riemann Integrable:
 * $ \displaystyle \int_{1}^{x} \dfrac{1}{t} \ \mathrm d t$ exists

Hence the result, since Riemann integrals are unique.