Question:
If \(\log_{10} 2,\ \log_{10}(2^x – 1),\ \log_{10}(2^x + 3)\) are in AP, then what is x equal to?
यदि \(\log_{10} 2,\ \log_{10}(2^x – 1),\ \log_{10}(2^x + 3)\) समांतर श्रेढ़ी (AP) में हैं, तो x किसके बराबर है?
Options:
- \(\log_5 2\)
- \(\log_2 5\)
- 0
- 1
- \(\log_5 2\)
- \(\log_2 5\)
- 0
- 1
Correct Answer: b)
Explanation:
Since the terms are in AP:
\(2\log_{10}(2^x-1) = \log_{10} 2 + \log_{10}(2^x+3) = \log_{10}[2(2^x+3)]\).
\(\implies (2^x-1)^2 = 2(2^x+3)\).
Let \(y = 2^x\): \((y-1)^2 = 2y+6 \implies y^2 – 4y – 5 = 0 \implies (y-5)(y+1) = 0\).
Since \(2^x > 0\), \(y = 5 \implies 2^x = 5 \implies x = \log_2 5\).
चूँकि पद समांतर श्रेढ़ी में हैं:
\(2\log_{10}(2^x-1) = \log_{10}[2(2^x+3)]\)।
\(\implies (2^x-1)^2 = 2(2^x+3)\)।
माना \(y = 2^x\): \(y^2 – 4y – 5 = 0 \implies (y-5)(y+1) = 0\)।
चूँकि \(2^x > 0\) होता है, अतः \(y = 5 \implies 2^x = 5 \implies x = \log_2 5\)।