Question

Find the number of real values of x satisfying the equation:
\[ \log_{2}(x^2 - 5x + 6) + \log_{1/2}(x - 2) = 3 \]

Hint:

Before combining anything, work out exactly which x-values make both log arguments positive — this domain restriction is what rules out any extra candidate later. Also remember that a log with base $1/2$ is just the negative of the same log with base $2$, so either one can be rewritten in terms of the other.

Answer 1

Correct Answer: 1

Before solving anything, pin down the domain: \(x^2-5x+6=(x-2)(x-3)\) must be positive, which happens when \(x<2\) or \(x>3\), and separately \(x-2\) must be positive, meaning \(x>2\). Overlapping these two requirements leaves only \(x>3\) as the allowed region, a single unbroken interval. Combining the two logarithms turns the equation into \(\log_{2}(x-3)=3\), a linear equation in \(x-3\) once the log is stripped away, and a linear equation of this type has at most one root, here \(x=11\), which indeed lies in \(x>3\). Since the domain is one connected interval and the reduced equation is linear (not quadratic or higher), there is no room for a second solution to appear or for the found root to be spurious, so the answer is \(\boxed{1}\) real value.