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Question

# The values of x satisfying xlog5x>5 lie in the interval

A
(0,)
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B
(0,15)(5,)
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C
(1,)
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D
(1,2)
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Solution

## The correct option is B (0,15)∪(5,∞)We have, xlog5x>5 Clearly, it is meaningful for x>0. Now, two cases arise: Case 1: When x>1 In this case, we have xlog5x>5 ⇒log5x>logx5[∵ax>N⇒x>logaN if a>1] ⇒log5x>1log5x ⇒(log5x−1)(log5x+1)>0 ⇒log5x−1>0⇒x>5⇒x∈(5,∞) Case 2: When 0<x<1 In this case, we have xlog5x>5 ⇒log5x<logx5[∵ax>N⇒x<logaN if 0<a<1] ⇒log5x<1log5x ⇒(log5x)2−1>0 (∵ log5x<0) ⇒(log5x−1)(log5x+1)>0 ⇒(log5x+1)<0⇒x<5−1⇒x∈(0,15) Hence, x∈(0,15)∪(5,∞)

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