In the attachment below you can see the definition of the likelihood function. Likelihood
1) Whilst the explanation of why the whole max likelihood method is viable for discrete distributions is clear to me, I struggle to persuade myself why this approach is valid for continuous cases. Why is just multiplying the densities appropriate?
2) I don't seem to understand the explanation why the $\log(L(\theta))$ may be used either. In the same attachment point 1 declares that "Strictly monotonic increasing functions $g$ preserve order..." Whilst this is the case for $\log(L(\theta))$ I don't understand what exactly it justifies. If I understand correctly, $L(\theta)$ can be decreasing OR decreasing. So I don't see the connection.
At the same time another explanation (from another source) sounded: "This is because the two functions, $\log(L(\theta))$ and $L(\theta)$ are monotonically related to each other so the same MLE estimate is obtained by maximizing either one." In other words, no matter if $L(\theta)$ is growing or decreasing the $\log(L(\theta))$ will do the same. Would you agree that this is a more appropriate explanation?