Reference Request: the product of exponents of the same base is equivalent to the base raised to summation of the exponents

Question

I am trying to find a reference and name for this property of big operators and exponents, but for the life of me I cannot find it anywhere.

$$\prod_{i=1}^{n} e^{x_i} = e^{\sum_{i=1}^n x_i}$$

Thank you for your help :)

Answer 1

Writing the product and summation terms explicitly, the expression is actually just the basic property of exponential functions

$$e^a e^b e^c ... = e^{a+b+c+...}$$

Exponentiation turns multiplicative problems into additive ones. I'm not sure if there is a specific name for this property.

Answer 2

This follows fundamentally from the definitions in Group Theory.

For a generic, qualifying Binary Operator *, on set U, $\alpha, \beta \in U$, we have some $c\in U$ where $c=\alpha * \beta$.

But what if you just multiply $\alpha$ by itself repeatedly?

Suppose you have $x=\alpha ^m=\alpha * \alpha * \alpha... $ m times, and $y=\alpha ^n$. Then $xy=(\alpha*\alpha..)$[m times] $* (\alpha * \alpha ...)$[n times] , so $\alpha$ will be multiplied by itself $(m+n)$ times allowing us to rewrite $xy$ as $xy=\alpha ^{m+n}$.

This property holds for even more complicated groups like Multiplication on the set of $2$ x $2$ Invertible Matrices.

Answer 3

If you like, one way to name this property is that "The reals $\mathbb{R}$ are an "exponential field." See this Wikipedia entry.