Question

The product of all positive real values of x satisfying the equation
`\sf {x}^{(16( log_(5) {x})^(3) - 68 log_(5)x) } = {5}^( - 16)` is

Answer 1

Take the base-5 logarithm on both sides and simplify.

`x^(16 \log^3_5(x) - 68 \log_5(x)) = 5^(-16)`

`\left(16 \log^3_5(x) - 68 \log_5(x)\right) \log_5(x) = -16`

`16 \log^4_5(x) - 68 \log^2_5(x) + 16 = 0`

`4 \log^4_5(x) - 17 \log^2_5(x) + 4 = 0`

Factorize the left side.

`\left(4 \log^2_5(x) - 1\right) \left(\log^2_5(x) - 4\right) = 0`

Then

`4 \log^2_5(x) - 1 = 0 \implies \log^2_5(x) = \frac14 \\\\ ~~~~~~~~ \implies \log_5(x) = \pm \frac12 \\\\ ~~~~~~~~ \implies x = \frac1{\sqrt5} \text{ or } x = \sqrt5`

or

`\log^2_5(x) - 4 = 0 \implies \log^2_5(x) = 4 \\\\ ~~~~~~~~ \implies \log_5(x) = \pm 2 \\\\ ~~~~~~~~ \implies x = \frac1{25} \text{ or } x = 25`

and the product of these solutions is of course 1.