Equivalence of Definitions of Real Exponential Function

Theorem
All the definitions of the exponential function are equivalent.


 * $(1): \quad y = \exp x \iff \ln y = x$:


 * $\ln y = \displaystyle \int_{t \mathop = 1}^{t \mathop = y} \frac 1 t \ \mathrm dt$


 * $(2): \quad \exp x = e^x$


 * $(3): \quad \exp x = \displaystyle \lim_{n \mathop \to \infty} \left({1 + \frac x n}\right)^n$


 * $(4): \quad \exp x = \displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$


 * $(5): \quad y = f \left({x}\right): y = \dfrac {\mathrm dy}{\mathrm dx}, f \left({0}\right) = 1$

1 implies 5
This proves that $y$ is a solution.

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

5 implies 1
To solve for $C$, put $\left({x_0, y_0}\right) = \left({0, 1}\right)$:

3 implies 4
From the Binomial Theorem:


 * $\displaystyle \left({1 + \frac x n}\right)^n = 1 + x + \frac{n \left({n-1}\right)x^2}{2! \ n^2} + \frac{n \left({n-1}\right)\left({n-2}\right)x^3}{3! \ n^3} + \cdots$


 * $ = \displaystyle \frac {x^0}{0!} + \frac {x^1}{1!} + \left({\frac {n-1} {n} }\right) \frac {x^2}{2!} + \left(\right) \frac {x^3}{3!} + \cdots$

From Power of Number Less Than One, this converges to:


 * $\displaystyle \exp x - \frac {x^0}{0!} + \frac {x^1}{1!} + \frac {x^2}{2!} + \frac {x^3}{3!} + \cdots = 0$

as $n \to +\infty$:

Compare Series of Power over Factorial Converges.

3 implies 5
The application of Derivative of Exponential at Zero isn't circular as the referenced proof does not depend on $D_x \exp x = \exp x$.

Also see

 * Equivalence of Definitions of Euler's Number
 * Equivalence of Logarithm Definitions