Equivalence of Definitions of Real Exponential Function/Inverse of Natural Logarithm equivalent to Differential Equation

Proof
==== Inverse of Natural Logarithm implies Solution of Differential Equation ====

Let $\exp x$ be the real function defined as the inverse of the natural logarithm:
 * $y = \exp x \iff x = \ln y$

Then:

This proves that $y$ is a solution of the differential equation.

It remains to be proven that $y$ fulfils the initial condition:

That is:

$\exp x$ is the particular solution of the differential equation:
 * $\dfrac {\d y} {\d x} = y$

satisfying the initial condition $\map f 0 = 1$.

==== Solution of Differential Equation implies Inverse of Natural Logarithm ====

Let $\exp x$ be the real function defined as the particular solution of the differential equation:
 * $\dfrac {\d y} {\d x} = y$

satisfying the initial condition $\map f = 1$.

Thus:

To solve for $C$, put $\tuple {x_0, y_0} = \tuple {0, 1}$:

That is:
 * $y = \exp x \iff x = \ln y$