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JEE Main 20248 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
8 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2024 · 8 April · Morning Shift · Mathematics · Quadratic Equation

The sum of all the solutions of the equation (8)^ 2 x -16...

The sum of all the solutions of the equation (8)2x−16⋅(8)x+48=0(8)^{2 x}-16 \cdot(8)^x+48=0 is :

Options

  1. A1+log⁡8(6)1+\log _8(6)
  2. B1+log⁡6(8)1+\log _6(8)
  3. Clog⁡8(6)\log _8(6)
  4. Dlog⁡8(4)\log _8(4)

Correct answer

A. 1+log⁡8(6)1+\log _8(6)

Step-by-step solution

(8)2x−16⋅(8)x+48=0 Put 8x=tt2−16+48=0⇒t=4 or t=12⇒8x=48x=12⇒x=log⁡8xx=log⁡812\begin{aligned} & (8)^{2 x}-16 \cdot(8)^x+48=0 \\ & \text{ Put } 8^{\mathrm{x}}=\mathrm{t} \\ & \mathrm{t}^2-16+48=0 \\ & \Rightarrow \mathrm{t}=4 \text{ or } \mathrm{t}=12 \\ & \Rightarrow 8^x=4 \quad 8^x=12 \\ & \Rightarrow \mathrm{x}=\log _8 \mathrm{x} \quad \mathrm{x}=\log _8 12 \\ & \end{aligned} sum of solution =log⁡84+log⁡812=\log _8 4+\log _8 12=log⁡848=log⁡8(6.8)=1+log⁡86\begin{aligned} & =\log _8 48=\log _8(6.8) \\ & =1+\log _8 6\end{aligned}$$
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