A random variable $X$ has a probability density function of $f(x) = \dfrac{\alpha}{x^{\alpha +1}}$, $x > 1$ and $\alpha > 0$. Show that the distribution of $Y = \log X$ is exponential with rate $\alpha$ using:
(i) ideas from moment generating functions
I reword the information in the question to get:
$Y = \log(X)$ into $E[e^{sY}]$ to get $E[e^{s \log X}]$ and simplify to $E[X^s]$.
Using the integral definition of an expectation (not sure if the range is correct):
$$\int_1^\infty x^s \cdot f(x) \space dx$$
This is where I do not know how to take it further. What I need from here, is to show that the distribution of $Y$ is exponential with rate $\alpha$.
\begin{align} \operatorname{E}(e^{sY}) = \operatorname{E}(X^s) = \int_1^\infty x^s \frac \alpha {x^{\alpha+1}} \, dx = \alpha \int_1^\infty x^{s-\alpha - 1} \, dx = \alpha \left[ \frac{x^{s-\alpha}}{s-\alpha} \right]_1^\infty = \frac \alpha {\alpha - s}. \end{align} This last equality holds if $s< \alpha,$ and that is enough. Ask yourself for which distribution this is a moment-generating function.
(You may need to cite a theorem saying if a distribution has the same moment-generating function as a certain other distribution, then they are the same distribution.)