Arithmetic Mean of Logarithmic Terms

CAT 2017 Slot 2 · QA · Hard · Exponents & Logarithms

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If log⁡(2a×3b×5c)\log(2^a \times 3^b \times 5^c) is the arithmetic mean of log⁡(22×33×5)\log(2^2 \times 3^3 \times 5), log⁡(26×3×57)\log(2^6 \times 3 \times 5^7), and log⁡(2×32×54)\log(2 \times 3^2 \times 5^4), then aa equals

Answer

3

Explanation

Arithmetic mean of three logs = 13[log⁡X+log⁡Y+log⁡Z]=log⁡(XYZ)1/3\frac{1}{3} [\log X + \log Y + \log Z] = \log(X Y Z)^{1/3}.

Product of arguments: (22×33×5)×(26×3×57)×(21×32×54)=22+6+1×33+1+2×51+7+4(2^2 \times 3^3 \times 5) \times (2^6 \times 3 \times 5^7) \times (2^1 \times 3^2 \times 5^4) = 2^{2+6+1} \times 3^{3+1+2} \times 5^{1+7+4} =29×36×512= 2^9 \times 3^6 \times 5^{12}

Taking cube root: (29×36×512)1/3=23×32×54(2^9 \times 3^6 \times 5^{12})^{1/3} = 2^3 \times 3^2 \times 5^4

Comparing with 2a×3b×5c2^a \times 3^b \times 5^c: a=3a = 3.

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