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The value of log(5)sqrt(5sqrt(5sqrt(5......

The value of `log_(5)sqrt(5sqrt(5sqrt(5...^(@))))+log((1)/(2)+((1)/(2))^(2)+((1)/(2))^(3)+...oo)` is ______.

A

1

B

25

C

10

D

20

Text Solution

Verified by Experts

The correct Answer is:
A
Let `x = sqrt(5sqrt(5sqrt(5 ... oo)))`
`x = sqrt(5x)`
`x^(2) = 5x`
`implies x = 5 (because x ne 0)`
`therefore log_(5) sqrt(5sqrt(5 ... oo)) = log_(5) 5 = 1`.
Consider `(1)/(2) + ((1)/(2))^(2) + ((1)/(2))^(3) + ... + oo`
Clearly it is a GP, `s_(oo) = (a)/(1-r)`
Here `a = (1)/(2)` and `r = (1)/(2)`
`=((1)/(2))/(1-(1)/(2))=1`
`therefore log ((1)/(2) + ((1)/(2))^(2) + ... ) = log 1 = 0`.
`therefore` The required value = 1 + 0 = 1.
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