If 6^{3-4x}4^{x+5}=8 (Given \log_{10}2=0.301 and \log_{10}3=0.477), then which one of the following is correct?
- A. 0 \lt x \lt 1
- B. 1 \lt x \lt 2 ✓
- C. 2 \lt x \lt 3
- D. 3 \lt x \lt 4
Correct Answer: B. 1 \lt x \lt 2
Explanation
Taking \log_{10} on both sides: (3-4x)(\log 2 + \log 3) + (x+5)(2\log 2) = 3\log 2. Substituting values: (3-4x)(0.778) + (2x+10)(0.301) = 0.903. This simplifies to 2.334 - 3.112x + 0.602x + 3.010 = 0.903, giving 2.51x = 4.441, so x \approx 1.76. Thus, 1 \lt x \lt 2.
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