how to take the natural log of a product ($\prod$)

The log of a product is the sum of logs of the things inside the product. So $$\ln L(\theta)=\sum_{i=1}^n \ln\left(\frac{1}{\theta}e^{-x_i/\theta}\right)=\sum_{i=1}^n \left(\ln\left(\frac{1}{\theta}\right)-\frac{x_i}{\theta}\right)$$

$$\ln{(L(\theta))} =\sum_{i=1} ^n \ln\left(\frac{1}{\theta}e^{-\frac{x_i}{\theta}}\right) $$

This is because of the product rule for logarithms, that says that $\log_a (BC) = \log_a (B) + \log_a (C)$.