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 question requires us to solve a logarithmic equation for the variable 'x'. The key steps involve simplifying the equation using logarithmic properties and ensuring that the final solution is valid by checking it against the domain of the logarithmic functions.
Step 2: Key Formula or Approach:
We will use the following fundamental properties of logarithms:
1. Domain Condition: For any logarithm \(\log_{b}(A)\), the argument A must be positive, i.e., \(A>0\).
2. Change of Base Property: A key identity is \(\log_{1/b}(A) = \log_{b^{-1}}(A) = -\log_{b}(A)\).
3. Quotient Rule: \(\log_{b}(M) - \log_{b}(N) = \log_{b}\left(\frac{M}{N}\right)\).
4. Exponential Form: The equation \(\log_{b}(A) = C\) is equivalent to \(A = b^C\).
Step 3: Detailed Explanation:
Part A: Determine the Domain of x
For the logarithms to be defined, their arguments must be greater than zero.
Condition 1: From \(\log_{2}(x^2 - 5x + 6)\): \[ x^2 - 5x + 6>0 \] \[ (x - 2)(x - 3)>0 \] This inequality holds true for \(x<2\) or \(x>3\).
Condition 2: From \(\log_{1/2}(x - 2)\): \[ x - 2>0 \] \[ x>2 \] For the equation to be valid, both conditions must be met. The intersection of (\(x<2\) or \(x>3\)) and (\(x>2\)) is \(x>3\). Therefore, any valid solution must be greater than 3.
Part B: Solve the Equation
First, simplify the term with base 1/2 using the change of base property: \[ \log_{1/2}(x - 2) = -\log_{2}(x - 2) \] Substitute this into the original equation: \[ \log_{2}(x^2 - 5x + 6) - \log_{2}(x - 2) = 3 \] Apply the quotient rule for logarithms: \[ \log_{2}\left(\frac{x^2 - 5x + 6}{x - 2}\right) = 3 \] Factor the numerator of the argument: \[ \log_{2}\left(\frac{(x - 2)(x - 3)}{x - 2}\right) = 3 \] Since the domain is \(x>3\), we know that \(x - 2 \neq 0\), so we can safely cancel the `(x-2)` terms: \[ \log_{2}(x - 3) = 3 \] Now, convert the logarithmic equation to its exponential form: \[ x - 3 = 2^3 \] \[ x - 3 = 8 \] \[ x = 11 \] Part C: Verify the Solution
The solution we found is \(x = 11\). We must check if it lies within the domain we established (\(x>3\)).
Since \(11>3\), the solution is valid.
Step 4: Final Answer:
The only real value of x that satisfies the equation is 11.