Problem 9 - Entrance Test

If log₂(x) + log₂(x - 2) = 3, what is the value of x?

A. 1B. 2C. 3D. 4E. 5

Correct: D

log₂(x) + log₂(x - 2) = log₂(x(x - 2)) = 3. Therefore, x(x - 2) = 2^3 = 8. x^2 - 2x = 8 => x^2 - 2x - 8 = 0 => (x - 4)(x + 2) = 0. So, x = 4 or x = -2. Since we can't take the logarithm of a negative number, x must be 4. Checking: log₂(4) + log₂(4 - 2) = log₂(4) + log₂(2) = 2 + 1 = 3.