Question

If A is the set of all xϵR such that x(log x)2−3 log x+1>1000, and A=(a,∞) then √10a will be ___. ( (Base of logx is 10).

Solution

A=(1000,∞) (log x)2−3 log x+1>logx 103=3 logx 10 If log10 x=t then we have t2−3t+1>3t or t3−3t2+t−3>0 or t(t2+1)−3(t2+1)>0 or (t2+1)(t−3)>0⇒t−3>0 as t2+1 is always + ive ∴t>3 or log10x>3 ∴x>103=1000∴xϵ(1000,∞)

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