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Question

If $$\displaystyle \log_{10}x=y$$, then the value of $$\displaystyle \log _{1000}x^{2}$$ in terms of $$y$$ is


A
y3
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B
y3
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C
2y3
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D
2y3
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Solution

The correct option is C $$\dfrac{2y}3$$
$$\log _{ 10 } x=y$$
$$ \Rightarrow \dfrac { 2\log { x }  }{ 3\log { 10 }  } =\dfrac { 2y }{ 3 }\left[\because \log_ba=\dfrac{\log a}{\log a} \right] $$
$$ \Rightarrow \dfrac { \log { x^2 }  }{ \log { 10^3 }  }[\because x\log a=\log a^x] =\dfrac { 2y }{ 3 }$$
$$ \Rightarrow \log _{ 1000 }{ { x }^{ 2 } } =\dfrac { 2y }{ 3 } $$

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