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# If the `log_2(sin x) = -1` , then what is `log_2(csc x)`

alejandrogalarce | Student, Undergraduate | Valedictorian

Posted May 8, 2013 at 6:01 PM via web

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If the `log_2(sin x) = -1` , then what is `log_2(csc x)`

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted May 8, 2013 at 6:11 PM (Answer #1)

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It is given that `log_2(sin x) = -1` . Use the following property of logarithm `log x^a = a*log x`

`log_2(cosec x)`

= `log_2(1/sin x)`

= `log_2((sin x)^-1)`

= `-1*log_2 sin x`

But `log_2 sin x = -1`

= -1*-1

= 1

The value of `log_2 cosec x = 1`

oldnick | Valedictorian

Posted May 8, 2013 at 10:39 PM (Answer #2)

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`log_2 cosec x=log_2(1/sin x)=-log_2 sinx = -(-1)=1`

pramodpandey | College Teacher | Valedictorian

Posted May 9, 2013 at 8:17 AM (Answer #3)

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log is function whose domain is positive real numbers and co domain is real numbers.

If `log_2 (sin(x))=-1`

`` By def of log

`2^(-1)=sin(x)`

`sin(x)=1/2`

`(1/(cosec(x)))=1/2`

`cosec(x)=2`

Thus

`log_2 (cosec(x))=log_2 2=1`  (by def . of log)

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