Multiple choice

If $x_1$ and $x_2$ are the roots of the equation $e^2.x^{\ln x}=x^3$ with $x_1>x_2$, then what is the relation between the roots?

  1. $x_1=2x_2$
  2. $x_1=(x_2)^2$
  3. $2x_1=(x_2)^2$
  4. $(x_1)^2=(x_2)^3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Taking the natural log of both sides: ln(e^2 * x^ln x) = ln(x^3) becomes 2 + (ln x)^2 = 3 ln x. Let u = ln x, then u^2 - 3u + 2 = 0. Roots are u = 1, 2. Thus ln x1 = 2 (x1 = e^2) and ln x2 = 1 (x2 = e). Therefore, x1 = (x2)^2.

AI explanation

Taking the natural logarithm of both sides of the equation e squared times x to the power of (natural log of x) equals x cubed yields natural log of (e squared) plus (natural log of x) squared equals 3 times natural log of x. Substituting y for the natural log of x gives the quadratic equation y squared minus 3y plus 2 equals 0, which factors into (y minus 2) times (y minus 1) equals 0. The roots are y equals 2 and y equals 1, meaning the natural log of x1 is 2 and the natural log of x2 is 1. This gives x1 equals e squared and x2 equals e, confirming the relation x1 equals (x2) squared.

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