The product of all solutions of the equation $e^{5\left(\log_{e} x\right)^{2}+3}=x^{8}, x>0$, is :

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Basic Mathematics

The product of all solutions of the equation

e^{5 (l o g_{e} x)^{2} + 3} = x^{8}, x > 0

, is :

Answer: (A)

Take natural log of both sides of

e^{5 (l n x)^{2} + 3} = x^{8}

5 (ln x)^{2} + 3 = 8 ln x .

Let

t = ln x

5 t^{2} - 8 t + 3 = 0.

By Vieta's formulas, the sum of the roots is

t_{1} + t_{2} = \frac{8}{5}

.

Since

t_{1} = ln x_{1}

and

t_{2} = ln x_{2}

ln (x_{1} x_{2}) = t_{1} + t_{2} = \frac{8}{5} ⟹ x_{1} x_{2} = e^{8/5} .

The product of all solutions is

e^{8/5}

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Single correct option