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The equation logₓ(16) + log₂ₓ(64) = 3 has which of the following solutions?

1. one irrational solution
2. no prime solution
3. two real solutions
4. one integral solution

Correct answer: two real solutions

Solution

Substituting t = log₂(x) reduces the equation to the quadratic 3t² - 7t - 6 = 0, which has two real roots t = 3 and t = -2/3, giving x = 8 and x = 2^(-2/3). Both are valid (positive, not 1), so there are two real solutions.

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