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: Understanding the Question 
The problem requires solving a logarithmic equation for the variable x. We will simplify the equation using logarithmic identities and ensure that the final solution satisfies the domain conditions of all logarithmic expressions.

Step 2: Key Formulae / Properties Used
 

• Domain Condition:
  For any logarithm \( \log_b(A) \), the argument must be positive: \( A > 0 \).
• Change of Base (Reciprocal Base):
  \( \log_{1/b}(A) = -\log_b(A) \)
• Quotient Rule:
  \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \)
• Exponential Form:
  If \( \log_b(A) = C \), then \( A = b^C \)

Step 3: Detailed Solution

Part A: Determine the Domain of x
For the logarithmic expressions to be defined, their arguments must be positive.

Condition 1: From \( \log_2(x^2 - 5x + 6) \)
\[ x^2 - 5x + 6 > 0 \] \[ (x - 2)(x - 3) > 0 \] This inequality holds for \( x < 2 \) or \( x > 3 \).

Condition 2: From \( \log_{1/2}(x - 2) \)
\[ x - 2 > 0 \] \[ x > 2 \]
Taking the intersection of both conditions: \[ (x < 2 \text{ or } x > 3) \cap (x > 2) = x > 3 \] Hence, the valid domain is \( x > 3 \).

Part B: Solve the Equation
First, convert the logarithm with base \( \frac{1}{2} \): \[ \log_{1/2}(x - 2) = -\log_2(x - 2) \] Substituting into the equation: \[ \log_2(x^2 - 5x + 6) - \log_2(x - 2) = 3 \] Apply the quotient rule: \[ \log_2\left(\frac{x^2 - 5x + 6}{x - 2}\right) = 3 \] Factor the numerator: \[ \log_2\left(\frac{(x - 2)(x - 3)}{x - 2}\right) = 3 \] Since \( x > 3 \), \( x - 2 \neq 0 \), so we cancel: \[ \log_2(x - 3) = 3 \] Convert to exponential form: \[ x - 3 = 2^3 \] \[ x - 3 = 8 \] \[ x = 11 \]
Part C: Verification
The obtained solution is \( x = 11 \).
Since \( 11 > 3 \), it satisfies the domain condition.

Step 4: Final Answer
The only real value of x that satisfies the equation is 11.