Distribution of Functions of Random Variables In general, how would one find the distribution of $f(X)$ where $X$ is a random variable? Or consider the inverse problem of finding the distribution of $X$ given the distribution of $f(X)$. For example, what is the distribution of $\max(X_1, X_2, X_3)$ if $X_1, X_2$ and $X_3$ have the same distribution? Likewise, if one is given the distribution of $ Y = \log X$, then the distribution of $X$ is deduced by looking at $\text{exp}(Y)$?

Qiaochu is right.  There isn't a magic wand.  That said, there is a set of common procedures that can be applied to certain kinds of transformations.  One of the most important is the cdf (cumulative distribution function) method that you are already aware of.  (It's the one used in your previous question.)  Another is to do a change of variables, which is like the method of substitution for evaluating integrals.  You can see that procedure and others for handling some of the more common types of transformations at this web site.  (Some of the other examples there include finding maxes and mins, sums, convolutions, and linear transformations.)
