Multiple choice

How many roots are possible for the equation log2 x2 + logx2 = 3?

  1. 3

  2. 2

  3. 1

  4. None
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let log2(x) = u. Then log2(x^2) = 2u and logx(2) = 1/u. Equation: 2u + 1/u = 3. 2u^2 - 3u + 1 = 0. (2u-1)(u-1) = 0. u = 1/2 or u = 1. log2(x) = 1/2 => x = sqrt(2). log2(x) = 1 => x = 2. Two roots are possible.

AI explanation

Using logarithm properties, the equation log2(x squared) plus logx(2) equals 3 can be rewritten as 2 log2(x) plus 1 / log2(x) equals 3. Let t equal log2(x), giving the quadratic equation 2t squared minus 3t plus 1 equals 0. Factoring this quadratic yields (2t - 1)(t - 1) equals 0, so t equals 1/2 and t equals 1. Solving for x gives x squared equals 2 and x squared equals 4, which produces the two distinct positive roots x equals root 2 and x equals 2. Therefore, there are 2 possible roots.

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