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24 Nov 2021, 07:33
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If \(2^x = 3\), then which of the following must be true?
A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)
B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)
C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)
D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)
E. \(1 \frac{5}{6} < x < 2\)
A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)
B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)
C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)
D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)
E. \(1 \frac{5}{6} < x < 2\)
24 Nov 2021, 07:33
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Official Solution:
If \(2^x = 3\), then which of the following must be true?
A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)
B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)
C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)
D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)
E. \(1 \frac{5}{6} < x < 2\)
Square: \(2^{2x} = 3^2=9\);
\((2^{2x} = 9) > 2^3\), so \(2x > 3\);
\(x>1 \frac{1}{2}\)
Cube: \(2^{3x} = 3^3=27\);
\((2^{3x} = 27) < 2^5\), so \(3x < 5\);
\(x < 1 \frac{2}{3}\).
So, \(1 \frac{1}{2} < x < 1 \frac{2}{3}\).
Answer: B
If \(2^x = 3\), then which of the following must be true?
A. \(1 \frac{1}{3} < x < 1 \frac{1}{2}\)
B. \(1 \frac{1}{2} < x < 1 \frac{2}{3}\)
C. \(1 \frac{2}{3} < x < 1 \frac{3}{4}\)
D. \(1 \frac{3}{4} < x < 1 \frac{5}{6}\)
E. \(1 \frac{5}{6} < x < 2\)
Square: \(2^{2x} = 3^2=9\);
\((2^{2x} = 9) > 2^3\), so \(2x > 3\);
\(x>1 \frac{1}{2}\)
Cube: \(2^{3x} = 3^3=27\);
\((2^{3x} = 27) < 2^5\), so \(3x < 5\);
\(x < 1 \frac{2}{3}\).
So, \(1 \frac{1}{2} < x < 1 \frac{2}{3}\).
Answer: B
General Discussion
10 Jun 2025, 21:47
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I like the solution - it’s helpful.
