how come this summation after produkt?

Question

I am really stuck in this step. I hope, the context does not matter here, so i didnot provide what this is about. I am trying to get ML-Estimator. but the problem is, as i see in my textbook, how they changed the produkt to summation, why it became $\sum$

$$L(a;X)=\prod_{i=1}^{n}\frac{1}{2}a \cdot exp(-a\cdot|x_i|) = \frac{1}{2^n}a^n\cdot exp(-a\sum_{i=1}^{n}|x_i|)$$

THANKS for help!

Answer 1

This is just a property of the exponential function. Indeed $\exp(a+b)=\exp(a)\cdot\exp(b)$ and so induction shows that $$ \prod_{i=1}^n\exp(a_i)=\exp\left(\sum_{i=1}^n a_i\right). $$ Use this with $a_i=-\vartheta|x_i|$.

Answer 2

$$\prod_{i=1}^n\left(a\cdot\mathrm e^{u_i}\right)=a^n\cdot\prod_{i=1}^n\mathrm e^{u_i}=a^n\cdot\exp\left(\sum_{i=1}^nu_i\right)$$