Question

If \( \log_3 2, \log_3 (2^x - 5), \log_3 (2^x - 7/2) \) are in arithmetic progression, then what is the value of \( x \)?

• A. 2
• B. 3
• C. 4
• D. 5

Hint:

For logarithmic APs, equate differences using logarithm properties and solve resulting equations.

Answer 1

The Correct Option is B

For \( \log_3 2, \log_3 (2^x - 5), \log_3 (2^x - 7/2) \) to be in AP, the difference between consecutive terms is equal: 
\[ \log_3 (2^x - 5) - \log_3 2 = \log_3 (2^x - 7/2) - \log_3 (2^x - 5) \] \[ \log_3 \left( \frac{2^x - 5}{2} \right) = \log_3 \left( \frac{2^x - 7/2}{2^x - 5} \right) \] \[ \frac{2^x - 5}{2} = \frac{2^x - 7/2}{2^x - 5} \] Cross-multiply: 
\[ (2^x - 5)^2 = 2 \left( 2^x - \frac{7}{2} \right) \] \[ 2^{2x} - 10 \cdot 2^x + 25 = 2 \cdot 2^x - 7 \] \[ 2^{2x} - 12 \cdot 2^x + 32 = 0 \] Let \( y = 2^x \): 
\[ y^2 - 12y + 32 = 0 \Rightarrow y = \frac{12 \pm \sqrt{144 - 128}}{2} = \frac{12 \pm 4}{2} = 8, 4 \] \[ 2^x = 8 \Rightarrow x = 3, \quad 2^x = 4 \Rightarrow x = 2 \] Verify: For \( x = 3 \): \( \log_3 2, \log_3 (8 - 5) = \log_3 3 = 1, \log_3 (8 - 7/2) = \log_3 4.5 \). Check AP condition numerically. 
Thus, the answer is 3.