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CDS 2025 · PYQ · Logarithms · medium

What is log₁₀ 2000 + log₁₀ 400 + 4 log₁₀ 25 + 5 log₁₀ 20 equal to?

  1. A.10
  2. B.16
  3. C.18✓ Correct
  4. D.20

Explanation

= log(2000) + log(400) + log(25⁴) + log(20⁵) = log(2000 × 400 × 25⁴ × 20⁵). Compute: 2000 = 2·10³, 400 = 4·10² = 2²·10², 25⁴ = (5²)⁴ = 5⁸, 20⁵ = (4·5)⁵ = 2¹⁰·5⁵. Product = 2·10³ × 2²·10² × 5⁸ × 2¹⁰·5⁵ = 2¹³ × 10⁵ × 5¹³ = (2·5)¹³ × 10⁵ = 10¹³ × 10⁵ = 10¹⁸. So log₁₀(10¹⁸) = 18.
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