What property is being used?

Question

I'm looking over a stats problem and there is a jump which I don't understand. It is:

$$\theta_1^{-n}(\prod_{i=1}^{n}x_i)^{1-\theta_1}=\theta_1^{-n}exp((\theta_1-1)(-\sum_{i=1}^{n}logx_i))$$

This question is about likelihood functions. We also know that $f(x|\theta)=\theta x^{\theta-1}$ for $x\epsilon(0,1)$ with $\theta>0$

Answer 1

Use the fact that for $x \in \mathbb{R}_{++},$ $$x = e^{\log{x}}.$$ Also $$\log{(ab)} = \log{(a)} + \log{(b)}$$ for $a,b \in \mathbb{R}_{++}$. Here $\mathbb{R}_{++}$ indicates strictly positive numbers.