Question

The product of all positive real values of $x$ satisfying the equation $x^{\left(16\left(\log _5 x\right)^3-68 \log _5 x\right)}=5^{-16}$ is _____.

Answer 1

Taking log5 on both sides
$(16(\log_5x)^3 -68(\log_5x))(\log_5x) = -16$
Let ( log5x)=t
16t4 - 68t2 + 16 = 0
4t4 + 16t2 - t2+4=0
(4t2 - 1)(t2-4) = 0
$t=±\frac{1}{2}\ or ±2$
So log5x = $±\frac{1}{2}\ or ±2$
$⇒ x=5^{\frac{1}{2}} , 5^{-\frac{1}{2}} , 5^2, 5^{-2}$
$\therefore$ Product = $(5)^{\frac{1}{2}-\frac{1}{2} +2-2}$
$= 5^0$
$= 1$

So, the correct answer is 1.