Question

Find the real value(s) of x that satisfy the equation:
\[ \log_{2}(x^2 - 5x + 6) + \log_{1/2}(x - 2) = 3 \]

Hint:

The first and most critical step in solving logarithmic equations is to determine the domain of the variable. This helps you to immediately discard any extraneous solutions you might find during your calculations.

Answer 1

Correct Answer: 11

Step 1: Rewrite \(\log_{1/2}(x-2)\) as \(-\log_2(x-2)\), so the equation becomes \(\log_2(x^2-5x+6)-\log_2(x-2)=3\).
Step 2: Combine into one logarithm and factor the quadratic: \[ \log_2\!\left(\frac{(x-2)(x-3)}{x-2}\right)=3 \implies \log_2(x-3)=3 \quad (x\neq2). \]
Step 3: Convert to exponential form: \(x-3=2^3=8 \implies x=11\).
Step 4 (verify): At \(x=11\): \(x^2-5x+6=72\), \(x-2=9\), both positive, and \(\log_2(72)-\log_2(9)=\log_2(8)=3\). Checks out.

\[ \boxed{x=11} \]