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Question

If A is the set of all xϵR such that x(log x)23 log x+1>1000, and A=(a,) then 10a will be ___. ( (Base of logx is 10).


Solution

A=(1000,)
(log x)23 log x+1>logx 103=3 logx 10
If log10 x=t then we have
t23t+1>3t
or t33t2+t3>0
or t(t2+1)3(t2+1)>0
or (t2+1)(t3)>0t3>0
as t2+1 is always + ive
t>3 or log10x>3
x>103=1000xϵ(1000,)

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