How do I take the natural log of the product $L(\theta) = \prod _{i=1}^n\left(\frac{1}{\theta \:}e^{-\frac{x_i}{\theta \:}}\right)$

Question

I have the likelihood function $L(\theta) = \prod _{i=1}^n\left(\frac{1}{\theta \:}e^{-\frac{x_i}{\theta \:}}\right)$. I'm trying to take the natural log, $\ln(L(\theta))$, but I'm not sure how this works with respect to $\prod$. Does anyone know what the process for this log is?

Answer 1

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

Answer 2

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

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