Question

If \( \log_2 x + \log_2 (x - 2) = 3 \), what is \( x \)?

• A. 2
• B. 3
• C. 4
• D. 6

Hint:

Combine logarithms using \( \log a + \log b = \log (a \cdot b) \), solve the resulting equation, and check domain.

Answer 1

The Correct Option is C

We need to solve for \( x \).
- Step 1: Apply logarithm property. \( \log_2 x + \log_2 (x - 2) = \log_2 [x (x - 2)] \).
- Step 2: Set up equation.
\[ \log_2 [x (x - 2)] = 3 \Rightarrow x (x - 2) = 2^3 = 8 \] - Step 3: Form quadratic equation.
\[ x^2 - 2x - 8 = 0 \] - Step 4: Solve quadrati(c) Use quadratic formula: \( x = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2} \).
\[ x = 4 \text{ or } x = -2 \] - Step 5: Check domain. Logarithms require positive arguments:
- For \( x = 4 \): \( x = 4>0 \), \( x - 2 = 2>0 \). Vali(d)
- For \( x = -2 \): \( x - 2 = -4<0 \). Invali(d)
- Step 6: Verify solution. For \( x = 4 \):
\[ \log_2 4 + \log_2 2 = 2 + 1 = 3 \] Correct.
- Step 7: Check options.
- (a) 2: \( \log_2 2 + \log_2 0 \). Invali(d)
- (b) 3: \( \log_2 3 + \log_2 1 \approx 1.58 + 0 \neq 3 \).
- (c) 4: Correct.
- (d) 6: \( \log_2 6 + \log_2 4 \approx 2.58 + 2 \neq 3 \).
Thus, the answer is c.