are these equivalent?

Question

I am trying to determine if these two expressions are equivalent:

$$\prod_{i=1}^n\frac{\log(\theta)}{\theta - 1} \theta^{x_i} = \frac{(\log(\theta))^n}{(\theta - 1)^n}\theta^{\sum_{i=1}^n x_i}$$

If they aren't what am I missing here?

Answer 1

Assuming that theta is not defined in terms of i, then indeed they are correct: $$ \prod_i^N{k\theta^{x_i}} = k^N\prod_i^N{\theta^{x_i}}=k^N\theta^{\sum_i^N{x_i}} $$

Answer 2

Note:

$$\prod_i^n\frac{log(\theta)}{\theta-1}\theta^{x_i}=\prod_i^n\frac{log(\theta)}{\theta-1}*\prod_i^n\theta^{x_i}=\left(\frac{log(\theta)}{\theta-1}\right)^n\theta^{\sum_i^n x_i}$$

Hope this is helpful!