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Question
Solve for x.
x + log10 (1 + 2x) = x log10 5 + log10 6
Solution
x + log10 (1 + 2x) = x log10 5 + log10 6
∴ x log10 10 + log10 (1 + 2x) = x log10 5 + log10 6 ...[∵ loga a = 1]
∴ log10 10x + log10 (1 + 2x) = log10 5x + log10 6
∴ log10 [10x (1 + 2x)] = log10 (6 × 5x)
∴ 10x (1 + 2x) = 6 × 5x
∴ 2x × 5x (1 + 2x) = 6 × 5x
∴ 2x (1 + 2x) = 6
Let 2x = a
∴ a.(1 + a) = 6
∴ a + a2 = 6
∴ a2 + a – 6 = 0
∴ (a + 3)(a – 2) = 0
∴ a + 3 = 0 or a – 2 = 0
∴ a = – 3 or a = 2
Since 2x = – 3, which is not possible
∴ 2x = 2 = 21
∴ x = 1
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