Multiple choice

The product of roots of the equation $\displaystyle \frac{\log_8(\frac{8}{x^2})}{(\log_8x)^2}=3$ will be

  1. $1$
  2. $\dfrac12$
  3. $\dfrac13$
  4. $\dfrac14$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let y = log_8(x). The equation is (log_8(8) - log_8(x^2)) / y^2 = 3. (1 - 2y) / y^2 = 3. 3y^2 + 2y - 1 = 0. (3y - 1)(y + 1) = 0. y = 1/3 or y = -1. x = 8^(1/3) = 2 or x = 8^-1 = 1/8. Product = 2 * 1/8 = 1/4.

AI explanation

Using logarithm properties, log_8(8/x^2) = log_8(8) - log_8(x^2) = 1 - 2log_8(x). Let y = log_8(x), which changes the left side of the equation to (1 - 2y) / y^2. Setting this equal to 3 gives 1 - 2y = 3y^2, or 3y^2 + 2y - 1 = 0. Factoring this quadratic gives (3y - 1)(y + 1) = 0, so the roots are y = 1/3 and y = -1. Re-substituting y = log_8(x) yields log_8(x) = 1/3 and log_8(x) = -1, giving the roots x = 2 and x = 1/8. The product of these two roots is 2 * 1/8 = 1/4.

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