Question

Solve for x.

x + log₁₀ (1 + 2^x) = x log₁₀ 5 + log₁₀ 6

Answer 1

x + log₁₀ (1 + 2^x) = x log₁₀ 5 + log₁₀ 6

∴ x log₁₀ 10 + log₁₀ (1 + 2^x) = x log₁₀ 5 + log₁₀ 6  ...[∵ log_a a = 1]

∴ log₁₀ 10^x + log₁₀ (1 + 2^x) = log₁₀ 5^x + log₁₀ 6

∴ log₁₀ [10^x (1 + 2^x)] = log₁₀ (6 × 5^x)

∴ 10^x (1 + 2^x) = 6 × 5^x

∴ 2^x × 5^x (1 + 2^x) = 6 × 5^x

∴ 2^x (1 + 2^x) = 6

Let 2^x = 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 2^x = – 3, which is not possible

∴ 2^x = 2 = 2¹

∴ x = 1