Question

What is the Exponential Property Involving Products?

Answer 1

Hello !

The exponential function `x \mapsto e^x` has a fundamental property involving products :

`\forall x,y \in \mathbb{R}, \quad e^{x+y} = e^x \times e^y`.

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

`e^{x_1+x_2+\ldots + x_n} = e^{x_1}\times e^{x_1} \times \ldots \times e^{x_n}`

for any numbers `x_1,\ldots,x_n`.

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