JEE Advanced · Mathematics · 1. Basic Math

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

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Answer & Solution

Correct Answer

(C) 1

Step-by-step Solution

Detailed explanation

Taking log to the base

5

on both sides

(16 {(\log_{5} x)}^{3} - 68 (\log_{5} x)) (\log_{5} x) = - 16

Let

(\log_{5} x) = t

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

\Rightarrow 4 t^{4} - 16 t^{2} - t^{2} + 4 = 0

\Rightarrow (4 t^{2} - 1) (t^{2} - 4) = 0

\Rightarrow t = \pm \frac{1}{2}, \pm 2

\log_{5} x = \pm \frac{1}{2}

\pm 2

\Rightarrow x = 5^{\frac{1}{2}}, 5^{\frac{- 1}{2}}, 5^{2}, 5^{- 2}

Hence, the product is

\sqrt{5} \times \frac{1}{\sqrt{5}} \times 25 \times \frac{1}{25} = 1

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The product of all positive real values of x satisfying the equation x16log5x3-68log5x=5-16 is

Step-by-step Solution

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