Question

Given log2 x=3/logxy 2 , express y in terms of x​

Answer 1

Answer:

             $\bold{y=\dfrac{\sqrt[3]{\bold x}}x=\bold \big x^{-\frac23} }$

Step-by-step explanation:

$\log_2x=\dfrac3{\log_{xy}2}\qquad\qquad\qquad\quad x>0\,,\ y>0\\\\\\\log_2x=3\cdot\dfrac1{\log_{xy}2}\\\\\\\log_2x=3\cdo\log_2(xy)\\\\\log_2x=\log_2(xy)^3\quad\iff\quad x=(xy)^3\\\\x=x^3y^3\\\\\dfrac{x}{x^3}=y^3\\\\y=\sqrt[\big3]{\dfrac{x}{x^3}}\\\\y=\dfrac{\sqrt[3]x}x$

{or:  $\dfrac{x}{x^3}=y^3\quad \implies\quad y=\left(\dfrac{x}{x^3}\right)^\frac13=\left(\big x^{-2}\right)^\frac13=\big x^{-\frac23}$}