Equivalence of Definitions of Real Exponential Function

Limit of Sequence implies Extension of Rational Exponential
=== Extension of Rational Exponential implies Solution of Differential Equation ===

Sum of Series equivalent to Solution of Differential Equation
=== Inverse of Natural Logarithm equivalent to Solution of Differential Equation ===

Also see

 * Equivalence of Definitions of Complex Exponential Function
 * Equivalence of Definitions of Euler's Number
 * Equivalence of Definitions of Real Natural Logarithm