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Question

Let f(x) be defined for all x>0 and be continuous,Let f(x) satisfy f(xy)=f(x)f(y) for all x,y, f(e)=1 Then

A
f(x) is bounded.
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B
f(1x)0asx0
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C
xf(1) as x(0)
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D
f(x)=logx
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Solution

The correct option is C f(x)=logx
Since,f(xy)=f(x)f(y) for all x,y
Clearly,the functionf(x)=alogx,where a is any real number.
Since,f(e)=aloge=1a=1,
f(x)=logx
Clearly,fis not bounded,
As x0,f(1x) and xf(x)0
So, the correct option is (D).

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