Assume $X_1, ..., X_n$ has a probability density function $$f(x|\theta) = \frac{\theta}{x^{\theta + 1}}$$
for $1 < x < \infty$ and $\theta > 1$.
I found that the MLE is $$\hat{\theta} = \frac{n}{\sum_{i=1}^n \text{log}(x_i)}$$
Now I have to find the distribution of $T = \sum_{i=1}^n \text{log}(x_i)$. I think a good way to do this is to start by writing $$f(x|\theta) = \theta e^{-\theta \log(x)}$$ and then consider $\log(\theta e^{-\theta \log(x)})$, but I do not see anything special here, even if I think there is something with an exponential distribution. Am I missing something?
Thank you!