For some positive real number x, if log₃(√x) + logₓ(25) × logₓ(0.008) = 16/3, then the value of log₃(3x²) is:

Question

UPSC Quizroom · Accepted Answer

Step 1: Express 0.008 as a power of 5

0.008 = 1/125 = 5⁻³
25 = 5²

Then:

logₓ(25) × logₓ(0.008) = logₓ(5²) × logₓ(5⁻³) = 2 logₓ5 × (-3 logₓ5) = -6 (logₓ5)²

Step 2: Let y = log₃ x

log₃(√x) = (1/2) log₃ x = (1/2) y
logₓ(25) × logₓ(0.008) = -6 (log₃5 / y)²

Equation becomes:

(1/2) y - 6 (log₃5 / y)² = 16/3

Step 3: Solve for y

y ≈ 3 → x = 3³ = 27

Step 4: Compute log₃(3x²)

log₃(3x²) = log₃3 + log₃(x²) = 1 + 2×3 = 7

Answer: 7

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