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 $x \ge 1$.)

Question

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 $x \ge 1$.)

amistre64 · Answer

hmm, will a substitution make life simpler

anonymous · Answer

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

zmudz · Answer

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.

zarkon · Answer

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