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Question

If $$(21.4)^a=(0.00214)^b=100$$, then the value of $$\left (\displaystyle \frac{1}{a}-\frac{1}{b} \right )$$, is


A
0
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B
1
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C
2
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D
4
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Solution

The correct option is C 2
Given that
$$(21.4)^a=(0.00214)^b=100$$
$$(21.4)^{ a }=100,(0.00214)^{ b }=100$$
Applying $$\log$$ on both the sides, we get
$$a=\log _{ 21.4 }{ 100,b=\log _{ 0.00214 }{ 100 }  } $$
$$\therefore \displaystyle \frac { 1 }{ a } -\frac { 1 }{ b } =\frac { 1 }{ \log _{ 21.4 }{ 100 }  } -\frac { 1 }{ \log _{ 0.00214 }{ 100 }  }, $$ $$\because$$ $$\left[\log _{ a }{ b } =\displaystyle \frac { 1 }{ \log _{ b }{ a }  }\right] $$
$$\displaystyle\Rightarrow  \frac { 1 }{ a } -\frac { 1 }{ b } =\log _{ 100 }{ 21.4 } -\log _{ 100 }{ 0.00214 } $$
$$\displaystyle\therefore  \frac { 1 }{ a } -\frac { 1 }{ b } =\log _{ 100 }{ \frac { 21.4 }{ 0.00214 }  } =\log _{ 100 }{ { 100 }^{ 2 } } =2\log _{ 100 }{ 100 } =2,$$
$$\because$$ $$\left[\log { a } -\log { b } =\log {\displaystyle  \frac { a }{ b }  } \right]$$

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