Question

The product of all positive real values of $x$ satisfying the equation $x^{\left(16\right({\log }_{5}x{)}^3-68{\log }_{5}x)}=5^{-16}$ is

• A. 1.0
• B. 1
• C. 1.00

Answer 1

Answer: (A), (B), (C)

Taking ${\log }_5$ on both sides
$\left(16\right({\log }_{5}x{)}^3-68\left({\log }_{5}x\right)\left)\right({\log }_{5}x)=-16$
Let $\left({\log }_{5}x\right)=t$
$16t^4-68t^2+16=0$
$4t^4-16t^2-t^2+4=0$
$(4t^2-1)(t^2-4)=0$
$t=\pm \frac{1}{2}$ or $\pm 2$
So ${\log }_{5}x=\pm \frac{1}{2}$ or $\pm 2$
$\Rightarrow x=5^{\frac{1}{2}},5^{\frac{-1}{2}},5^2,5^{-2}$
$\therefore$ Product $=(5{)}^{\frac{1}{2}-\frac{1}{2}+2-2}=5^0=1$