Question 4 - Jee advanced Math 2022 P2 Questions with Solutions

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 _____.

Sol : 

Let \(t = \log_{5}{x}\).

\(\implies x = 5^{t}\)

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

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

\(\implies t(16t^{3} - 68t) = -16\)

\(\implies 16t^{4} - 68t^{2} + 16 = 0\)

Let \(p = t^{2}\).

\(\implies 4p^{2} - 17p + 4 = 0\)

\(\implies 4p^{2} - 16p - p + 4 = 0\)

\(\implies 4p(p - 4) - 1(p - 4) = 0\)

\(\implies p = 4\) and \(p = \frac{1}{4}\)

\(\implies t = \pm 2\) and \(t = \pm \frac{1}{2}\)

\(\implies x = 5^{2} ; 5^{-2} ; 5^{\frac{1}{2}} ; 5^{-\frac{1}{2}}\)

All will yield positive value answers.

\(\implies\) product of all positive values of \(x\) 

         \( = 5^{2} \times 5^{-2} \times 5^{\frac{1}{2}} \times 5^{-\frac{1}{2}}\)

        \( = \frac{5^{2}}{5^{2}} \times \frac{5^{\frac{1}{2}}}{5^{\frac{1}{2}}}\)

         \( = 1\)


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