If log10 2, log10(2x - 1), and log10(2x + 3) are three consecutive terms of an arithmetic progression, which of the following is true?
- x = log2 5
- x = log2 3
- x = log5 2
- None of these
Explanation
Given that log10 2, log10(2x - 1), and log10(2x + 3) are in AP:
2 * log10(2x - 1) = log10 2 + log10(2x + 3)
log10(2x - 1)^2 = log10(2 * (2x + 3))
(2x - 1)^2 = 2 * (2x + 3)
4x^2 - 4x + 1 = 4x + 6
4x^2 - 8x - 5 = 0
(2x + 1)(2x - 5) = 0
2x = 5 or 2x = -1 ( rejected since 2x > 0 for log(2x-1) and log(2x+3) to be defined)
2x = 5
x = log2 5
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