The product of all positive real values of x satisfying the...

Taking log to the base $5$ on both sides

\left(16{\left(\log_{5}x\right)}^{3} - 68\left(\log_{5}x\right)\right)\left(\log_{5}x\right) = - 16$</p><p>Let \left(\log_{5}x\right) = t$

$16{t}^{4} - 68{t}^{2} + 16 = 0$

$⇒ 4{t}^{4} - 16{t}^{2} - {t}^{2} + 4 = 0$

$⇒ \left(4{t}^{2} - 1\right)\left({t}^{2} - 4\right) = 0$

$⇒ t = \pm \dfrac{1}{2}, \pm 2$

So $\log_{5}x = \pm \dfrac{1}{2}$ or $\pm 2$

$⇒ x = 5\dfrac{1}{2}, 5\dfrac{- 1}{2}, {5}^{2}, {5}^{- 2}$

Hence, the product is $\sqrt{5} \times \dfrac{1}{\sqrt{5}} \times 25 \times \dfrac{1}{25} = 1$