How did they arrive to 1/x?
I know that the conditional probability distribution is equal to: f(x,y)/ f_x(x) < where f_x(x) is the marginal distribution of X.
The probability density function for a "uniform probability" is a constant. In particular, the probability density function for the uniform probability on an interval, (a, b), is one over the length of the interval, $\displaystyle \frac{1}{b- a}$ Here, that interval is (0, x).