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Basic of Logarithms
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The product of all positive real values of $x$ satisfying the equation $x^{\left(16\left(\log _5 x\right)^3-68 \log _5 x\right)}=5^{-16}$is. . . . .
A
$0$
B
$1$
C
$4$
D
$5$
(IIT-2022)
Solution
$x^{16\left(\log _5 x\right)^3-68 \log _5 x}=5^{-16}$
Take log to the base 5 on both sides and put $\log _5 x=t$
$16 t^4-68 t^2+16=0$
$\Rightarrow 4 t^4-17 t^2+4=0\left\{\begin{array}{l}t_1 \\ t_2 \\ t_3 \\ t_4\end{array}\right.$
$t_1+t_2+t_3+t_4=0$
$\log _5 x_1+\log _5 x_2+\log _5 x_3+\log _5 x_4=0$
$x_1 x_2 x_3 x_4=1$
Standard 11
Mathematics
