Suppose, log 3 x = log 12 y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log 6 G is equal to:

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a — Easy Convert each log statement into an exponential to get $x$ and $y$. The geometric mean $G=\sqrt{xy}$ turns the product $3^a\cdot 12^a$ into $6^a$, so $\log_6 G=a$ immediately. 1 Solve for $x$ and $y$. Rewrite each log as a power: $$\begin{aligned} &\log_3 x=a\Rightarrow x=3^a\ &\log_{12} y=a\Rightarrow y=12^a \end{aligned}$$ 2 Geometric mean. $G=\sqrt{xy}$, and $3	imes 12=36=6^2$: $$\begin{aligned} &xy=3^a\cdot 12^a=36^a=(6^2)^a=6^{2a}\ &\Rightarrow\ G=\sqrt{xy}=6^{a} \end{aligned}$$ 3 Ta

CAT 2017 Slot 1 — QA Question 21

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