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log 10 log 2 log 2/pi (tan -1 x) -1

log10log2log2/pi(tan-1x)-1

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1 Answers

askIITianexpert IITDelhi
8 Points
12 years ago

Every base in above expression is +ve & not equal to 1 so that part of the requirement is fulfilled.

For tan-1x to be defined x could take any real value.

For (tan-1x)-1 to be defined x could take any real value except where tan-1x=0 ;i.e. x ≠0.

For log2/pi(tan-1x)-1 to be defined (tan-1x)-1 >0 ;i.e. x>0.

For log2log2/pi(tan-1x)-1  to be defined  log2/pi(tan-1x)-1 >0;or,(tan-1x)-1 >1;or, tan-1x<1;or,x<tan(1 radian).

At last for log10log2log2/pi(tan-1x)-1 to be defined log2log2/pi(tan-1x)-1 >0;or, log2/pi(tan-1x)-1 >1

or, (tan-1x)-1 <2/pi                 ....[coz' (2/pi)<1]

or, tan-1x>pi/2 which couldn't be possible for any real x.    ---[-¶/2≤tan-1x¶/2;principal values.]

Hence the given expression is not defined for any x or it's domain is a null set.

This could be easily deducted by doing last step first & such kind of intution could be mastered by good practice.I just give these steps for the thoroughness.

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