Multiple choice

If $(log_{10} x)^2 - 4 |log_{10} x| + 3 = 0$, the product of roots of the equation is :

  1. $3$
  2. $10^4$
  3. $10^6$
  4. $1$
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D Correct answer
Explanation

Let y = log10(x). The equation is y^2 - 4|y| + 3 = 0. Case 1: y >= 0, y^2 - 4y + 3 = 0 => (y-1)(y-3) = 0, so y=1 or y=3. x = 10^1 or 10^3. Case 2: y < 0, y^2 + 4y + 3 = 0 => (y+1)(y+3) = 0, so y=-1 or y=-3. x = 10^-1 or 10^-3. Product of roots = 10^1 * 10^3 * 10^-1 * 10^-3 = 10^0 = 1.

AI explanation

Let y = |log10 x|, which transforms the equation into the quadratic y^2 - 4y + 3 = 0. Factoring gives (y - 1)(y - 3) = 0, so the values for y are 1 and 3. Since y = |log10 x|, we get log10 x = 1, log10 x = -1, log10 x = 3, and log10 x = -3, yielding the roots 10, 0.1, 1000, and 0.001. The product of these roots is 10 * 0.1 * 1000 * 0.001, which equals 1.

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