Question

If ${2x}^4=y^4+z^4$, $xyz=8$ and ${log}_{y}x,{log}_{z}y,{log}_{x}z$ are in $G.P.$, then find the value of $x,y,z$ belonging to $R$.

Answer 1

${\left({log}_{z}y\right)}^2=\left({log}_{y}x\right)\left({log}_{x}z\right)$
or ${\left(\frac{logy}{logz}\right)}^2=\frac{logx}{logy},\frac{logz}{logx}$
or ${\left(logy\right)}^3={\left(logz\right)}^3\therefore y=z.$
$\therefore {2x}^4=2y^4\therefore x=y=z$
Also $xyz=8\Rightarrow x^3=8$
$\therefore x=2=y=z.$