Finding MLE of a distribution density, and derive a new MLE based off of the parameter $\theta$

Question

Given a distribution with density $$f(x)=\frac{x}{\theta^2}\exp(\frac{-x}{\theta})$$ How do I find the Maximum Likelihood Estimator of $\log(θ +7)$ ?

I have found the MLE of $\theta$ as $$\hat\theta=\frac{\bar{X}}{2}$$ with the four steps of

1. Likelihood Function
2. Log Likelihood function
3. Score equation (Equating the log Likelihood function to zero)
4. Solving the Score equation

but I have no idea how to proceed. This is the first time I'm posting a question here, so any feedback is appreciated.

Answer 1

Once you have the MLE $\hat{\theta}$ of $\theta$, the MLE of $f(\theta)$ is $f(\hat{\theta})$, since in both cases we're finding the point in parameter space that maximises the empirical data's likelihood.