Multiple choice

The number of roots of the equation $x^{\log_3x}\sqrt{x}=1$ is/are

  1. $1$
  2. $2$
  3. $3$
  4. $0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

x^(log3(x)) * x^(1/2) = 1. Taking log base 3: log3(x) * log3(x) + 1/2 * log3(x) = 0. Let u = log3(x). u^2 + 0.5u = 0. u(u + 0.5) = 0. u = 0 or u = -0.5. If u = 0, x = 3^0 = 1. If u = -0.5, x = 3^(-0.5) = 1/sqrt(3). Two roots.

AI explanation

Take the logarithm base 3 of both sides to get log base 3 of x times the quantity (log base 3 of x plus one half) equals 0. This yields two equations: log base 3 of x equals 0 or log base 3 of x equals negative one half. Solving these gives x equals 1 and x equals 1 divided by the square root of 3, which are both positive and valid. The number of roots is 2.

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