What is the number of solutions of \log_{4}(x-1)=\log_{2}(x-3)?
- A. Zero
- B. One ✓
- C. Two
- D. Three
Correct Answer: B. One
Explanation
\log_4(x-1) = \log_{2^2}(x-1) = \frac{1}{2} \log_2(x-1). The equation becomes \log_2(x-1) = 2 \log_2(x-3) = \log_2((x-3)^2). Thus, x-1 = x^2-6x+9 \implies x^2-7x+10=0, giving roots x=2, 5. Since \log_2(x-3) requires x \gt 3, x=2 is rejected. The <strong>ONLY</strong> solution is x=5.
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