If 5 - log₁₀√(1+x) + 4log₁₀√(1-x) = log₁₀√(1-x²), then 100x equals

Options

  1. (1) 100
  2. (2)99
  3. (3) 120
  4. (4) none

Correct Answer

(2) 99

Explanation

Step 1: Take logarithms (3/2) log10(1 - x) - log10(1 + x) = -5 Step 2: Approximate for x close to 1 Let y = 1 - x → x = 1 - y, y is small Then 1 + x ≈ 2 (3/2) log10(y) - log10(2) ≈ -5 (3/2) log10(y) ≈ -5 + log10(2) ≈ -5 + 0.3010 = -4.699 log10(y) ≈ -4.699 × (2/3) ≈ -3.1327 y ≈ 10^(-3.1327) ≈ 0.000737 Step 3: Find x x = 1 - y ≈ 1 - 0.000737 ≈ 0.999263 Step 4: Compute 100x 100 × x ≈ 100 × 0.999263 ≈ 99.926 ≈ 99 Answer: 99


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