Solve for x: log₂(x) + log₂(x - 2) = 3

Question

Accepted Answer

x = 4. The correct answer is x = 4. Using the product rule for logarithms: log₂(x(x - 2)) = 3. Converting to exponential form: x(x - 2) = 2³ = 8. Expanding: x² - 2x - 8 = 0, factoring: (x - 4)(x + 2) = 0, giving x = 4 or x = -2. However, x = -2 is extraneous because log₂(-2) is undefined. Only x = 4 is valid.

The answer x = 4 or x = -2 includes the extraneous solution. While x = -2 satisfies the quadratic equation, it does not satisfy the original logarithmic equation because logarithms require positive arguments. The answer x = 8 may result from incorrectly setting x = 2³ without properly combining the logarithms first. The answer x = 2 does not satisfy the equation: log₂(2) + log₂(0) is undefined since log₂(0) does not exist.

Solving logarithmic equations on the CLEP exam requires combining logarithms using properties, converting to exponential form, and critically, checking for extraneous solutions. Logarithmic functions are only defined for positive arguments, so any solution that makes an argument zero or negative must be discarded. This extraneous solution check is one of the most commonly tested aspects of logarithmic equations.

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