# What is the Exponential Property Involving Products?

Feb 18, 2015

Hello !

The exponential function $x \setminus \mapsto {e}^{x}$ has a fundamental property involving products :

$\setminus \forall x , y \setminus \in \setminus m a t h \boldsymbol{R} , \setminus \quad {e}^{x + y} = {e}^{x} \setminus \times {e}^{y}$.

So, exponential function transforms sums into products. Of course, you can write,

${e}^{{x}_{1} + {x}_{2} + \setminus \ldots + {x}_{n}} = {e}^{{x}_{1}} \setminus \times {e}^{{x}_{1}} \setminus \times \setminus \ldots \setminus \times {e}^{{x}_{n}}$

for any numbers ${x}_{1} , \setminus \ldots , {x}_{n}$.

Note that there exists other exponential functions : $x \setminus \mapsto {10}^{x}$, $x \setminus \mapsto {2}^{x}$, ..., $x \setminus \mapsto {a}^{x}$ for any positive real $a$. All of them have the same property involving products.