Question 144
Let $f:(0, \infty) \rightarrow (0, \infty)$ be a non-constant function with the property
$$
f\left(x^{y}\right) = (f(x))^{f(y)}
$$
for all $x, y \gt 0$. Show that
$$
f(xy) = f(x) f(y) \text{ and } f(x+y) = f(x) + f(y)
$$
for all $x, y \gt 0$.
$$
f\left(x^{y}\right) = (f(x))^{f(y)}
$$
for all $x, y \gt 0$. Show that
$$
f(xy) = f(x) f(y) \text{ and } f(x+y) = f(x) + f(y)
$$
for all $x, y \gt 0$.