Problem Dated 30.05.21

Algebra Level 3

If there exists a function $f(x)$ such that it always takes some positive value and there exists two positive integers $p$ and $q$ such that

$f(xf(y)) = x^p y^q \quad \forall x, y \in \mathbb R^+$

Find the value of $( \log_{q} {p})^{3}.$

1 solution

Baibhab Chakraborty
May 30, 2021

Here by putting $xf(y) = 1$ , we get

$f(1) = {x}^{p}.{y}^{q}$

$f(1) = \frac{{y}^{q}}{({f(y)})^{p}}$

$f(1) = \frac{1}{({f(1)})^{p}}$ [by putting $y=1$ ]

$(f(1))^{p+1} =1$ $\implies f(1) = 1$ ..............(1)

$f(xf(1)) = f(x) = {x}^{p}$ ..............(2)

$f(xf(y)) = {x}^{p}.(f(y))^{p} = {x}^{p}.{y}^{{p}^{2}} ={x}^{p}.{y}^{q}$ (given)

$q = p^{2}$

Thus $(log_q{p})^{3} = \frac{1}{8} = 0.125$

Hope this helps.:)