Logarithmic Relation and Geometric Mean

CAT 2017 Slot 1 · QA · Hard · Exponents and Logarithms

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Suppose, log⁡3x=log⁡12y=a\log_3 x = \log_{12} y = a, where x,yx, y are positive numbers. If GG is the geometric mean of xx and yy, then log⁡6G\log_6 G is equal to:

  1. A.

    a\sqrt{a}

  2. B.

    2a2a

  3. C.

    a2\frac{a}{2}

  4. D.

    aa

Answer

D

Explanation

x=3ax = 3^a and y=12ay = 12^a. Geometric mean G=xy=3a⋅12a=36a=6aG = \sqrt{xy} = \sqrt{3^a \cdot 12^a} = \sqrt{36^a} = 6^a. log⁡6G=log⁡6(6a)=a\log_6 G = \log_6 (6^a) = a.

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