One should mention that this is an example of a Pareto distribution, named after the Italian economist who used them to model distribution of incomes and of wealth, and they are alleged to model many phenomena (e.g., it has been alleged that 20% of books in a library account for 80% of circulation; you can make it an exercise to show that that corresponds to $\theta=\log_4 5$).
$ \int_1^\infty (\log x) \frac{\theta}{x^{\theta+1}}\,dx =\int_1^\infty (\log x) \frac{\theta}{x^\theta}\frac{dx}{x} = \int_0^\infty y \frac{\theta}{e^{y\theta}} \,dy = \int_0^\infty y e^{-\theta y}\,(\theta\,dy) $ $ = \frac 1 \theta \int_0^\infty (\theta y) e^{-\theta y} \, (\theta\,dy) = \frac 1 \theta \int_0^\infty u e^{-u}\,du = \frac 1 \theta. $
The last step elides an integration by parts: $ \int u e^{-u}\,du = \int u \, dv = uv - \int v\,du = ue^{-u} - \int -e^{-u}\,du = ue^{-u}+e^{-u}+\text{constant}. $ As $u\to\infty$, L'Hopital's rule shows that this approaches $0$. When $u=0$, this equals $1$.