If log_ 10 2, log_ 10 (2^x - 1), and log_ 10 (2^x + 3) are three consecutive terms of an arithmetic progression, which of the following is the value of x?

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Basic Mathematics QUESTION #7662

Question 1

If \(\log_{10} 2\), \(\log_{10}(2^x - 1)\), and \(\log_{10}(2^x + 3)\) are three consecutive terms of an arithmetic progression, which of the following is the value of \(x\)?

\(x = \log_2 2\)
\(x = \log_2 5\)✔️
\(x = \log_2 3\)
None of these

Correct Answer Explanation

For three terms to be in AP, \(2 \cdot \log_{10}(2^x - 1) = \log_{10} 2 + \log_{10}(2^x + 3)\). This simplifies to \(\log_{10}(2^x - 1)^2 = \log_{10}[2(2^x + 3)]\). Therefore, \((2^x - 1)^2 = 2(2^x + 3)\). Expanding: \(2^{2x} - 2 \cdot 2^x + 1 = 2 \cdot 2^x + 6\). Simplifying: \(2^{2x} - 4 \cdot 2^x - 5 = 0\). Let \(y = 2^x\): \(y^2 - 4y - 5 = 0\), giving \((y-5)(y+1) = 0\). Since \(y > 0\), \(y = 5\), so \(2^x = 5\) and \(x = \log_2 5\).

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