## zmudz one year ago Suppose that \(f(x)\) and \(g(x)\) are functions which satisfy \(f(g(x)) = x^2\) and \(g(f(x)) = x^3\) for all \(x \ge 1\). If \(g(16) = 16\), then compute \(\log_2 g(4)\). (You may assume that \(f(x) \ge 1\) and \(g(x) \ge 1\) for all \(x \ge 1\).)

1. amistre64

hmm, will a substitution make life simpler

2. anonymous

I don't see the link between the given information and the question. :( if g(16) =16, then g(4) =4, and \(log_2 g(4) = log_2 4 =2\)

3. zmudz

I already tried 2, it's not 2. But a hint says that I should find g(4) first. I just don't know how.

4. Zarkon

\[f(g(4))=4^2=16\] \[g(f(g(4)))=g(16)=16\] \[g(f(g(4)))=[g(4)]^3\] so \[[g(4)]^3=16\] \[\cdots\]