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If log(3) 2, log(3) ( 2^(x) - 5) and log...

If `log_(3) 2, log_(3) ( 2^(x) - 5)` and `log_(3) ( 2^(x) - ( 7)/( 2))` are in A.P., then x is equal to `:`

A

2

B

3

C

4

D

`2,3`

Text Solution

Verified by Experts

The correct Answer is:
B
`:. log_(3)2,log_(3)(2^(x)-5)` and `log_(3)(2^(x)-(7)/(2))" are in AP. " " " "……..(i)"`
For defined, `2^(x)-5gt0` and `2^(x)-(7)/(2)gt0`
`:. 2^(x)gt5 " " "…(ii)"`
From Eq.(i),`2,2^(x)-5,2^(x)-(7)/(2)` are in GP. `:." " (2^(x)-5)^(2)=2*(2^(x)-(7)/(2))`
`implies 2^(2x)-12*2^(x)+32=0`
`implies (2^(x)-8)(2^(x)-4)=0`
`:." " 2^(x)=8,4`
`implies 2^(x)=8=2^(3),2^((x) ne4 " " [" from Eq. (ii) "]`.
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