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If a gt 1, x gt 0 and 2^(log(a)(2x))=5^(...

If `a gt 1, x gt 0` and `2^(log_(a)(2x))=5^(log_(a)(5x))`, then x is equal to

A

`(1)/(10)`

B

`(1)/(5)`

C

`(1)/(2)`

D

1

Text Solution

Verified by Experts

The correct Answer is:
A
We having `2^(log_(a)(2x))=5^(log_(a)(5x))`
Taking log on both sides, we get
`log_(a)(2x).log 2=log_(a)(5x).log 5`
`rArr ((log 2+log x))/(log a)log 2=((log 5+ log x))/(log a)log 5`
`rArr (log 2)^(2)+log x log 2=(log 5)^(2)+(log x)log 5`
`rArr log x(log 2-log 5)=(log 5)^(2)-(log 2)^(2)`
`rArr -log x=log 5+log 2=log 10`
`rArr x=(1)/(10)`
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