Multiple choice

If $\displaystyle a> 1$ is a real number, then the number of roots of the equation $\displaystyle a^{2\log_{2}x}= 5+4x^{\log_{2}a}$ is/are

  1. $2$

  2. infinite
  3. $0$
  4. $1$
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D Correct answer
Explanation

Using the logarithmic identity x^(log_b(a)) = a^(log_b(x)), we can substitute t = a^(log_2(x)) to rewrite the equation as t^2 - 4t - 5 = 0. Solving this quadratic equation gives t = 5 or t = -1. Since t = a^(log_2(x)) must be positive, we only have t = 5, which yields exactly one unique solution for x.

AI explanation

Using the properties of logarithms, we can rewrite the given equation a^(2 log2 x) = 5 + 4x^(log2 a) as x^(2 log2 a) - 4x^(log2 a) - 5 = 0. Letting y = x^(log2 a), the equation becomes a quadratic y^2 - 4y - 5 = 0, which factors as (y - 5)(y + 1) = 0. Since x is positive, y must be positive, leaving y = 5 as the only valid root. The equation x^(log2 a) = 5 has exactly one valid solution for any given base a > 1, so the number of roots is 1.

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