Question

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

Answer 1

`\bold{y=\frac{\sqrt[3]{\bold x}}x=\bold \big x^(-\frac23) }`

Explanation:

`\log_2x=\frac3{\log_(xy)2}\qquad\qquad\qquad\quad x>0\,,\ y>0\\\\\\\log_2x=3\cdot\frac1{\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\\\\(x)/(x^3)=y^3\\\\y=\sqrt[\big3]{(x)/(x^3)}\\\\y=\frac{\sqrt[3]x}x`

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