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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>Nx>logaN if a>1]
log5x>1log5x
(log5x1)(log5x+1)>0
log5x1>0x>5x(5,)

Case 2: When 0<x<1
In this case, we have
xlog5x>5
log5x<logx5[ax>Nx<logaN if 0<a<1]
log5x<1log5x
(log5x)21>0 ( log5x<0)
(log5x1)(log5x+1)>0
(log5x+1)<0x<51x(0,15)

Hence, x(0,15)(5,)


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