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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Basics of Mathematics

JEE Advanced · Mathematics · Basics of Mathematics

If x_1 and x_2 are the solutions of the equation 5^ ( _5 x)^2 +x^...

If x1x_1 and x2x_2 are the solutions of the equation 5(log⁡5x)2+xlog⁡5x=12505^{\left(\log _5 x\right)^2}+x^{\log _5 x}=1250, then x1⋅x2x_1 \cdot x_2 is equal to:

Correct answer

1

Step-by-step solution

5(log⁡5x)2=xlog⁡5x=aa+a=1250⇒a=625=xlog⁡5x(log⁡5x)2=log⁡5625=4log⁡5x=2 or −2⇒x=−25 or 125\begin{aligned} 5^{\left(\log _5 x\right)^2} & =x^{\log _5 x}=a \\ a+a & =1250 \Rightarrow a=625=x^{\log _5 x} \\ \left(\log _5 x\right)^2 & =\log _5 625=4 \\ \log _5 x & =2 \text { or }-2 \Rightarrow x=-25 \text { or } \frac{1}{25}\end{aligned}
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