Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 9 / a / iv / Solution

The logarithm is strictly increasing, so $\mathbb{P}(Y\le y)=\mathbb{P}(X\le e^y)$. Since $X>1$ almost surely,Thus the logarithm of a Pareto random variable has an exponential distribution with rate $c$. Its probability density function value at the single endpoint can be chosen arbitrarily without changing the law.