Question

Let $\alpha =3^{{\log }_{4}5}-5^{{\log }_{4}3}+2.$ If $p$ and $q$ are the roots of the equation ${\log }_{\alpha }x+{\log }_x\alpha =\frac{10}{3},$ then the value of $p^3+q^3$ is

Answer 1

$\alpha =3^{{\log }_{4}5}-5^{{\log }_{4}3}+2$
$=5^{{\log }_{4}3}-5^{{\log }_{4}3}+2=2$

Now, ${\log }_{2}x+{\log }_{x}2=\frac{10}{3}$
Let ${\log }_{2}x=t$
$\Rightarrow t+\frac{1}{t}=\frac{10}{3}\Rightarrow 3t^2-10t+3=0$
$\Rightarrow t=3$ and $t=\frac{1}{3}$
$\Rightarrow {\log }_{2}x=3$ and ${\log }_{2}x=\frac{1}{3}$
$\Rightarrow x=2^3$ and $x=2^{1/3}$
$p=8$ and $q=2^{1/3}$
$\therefore p^3+q^3=512+2=514$