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
B
1
C
2
D
4

Solution

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

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