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Question

If f(x) is a function such that f(xy)=f(x)+f(y) and f(2)=1 then f(x)

A
x2
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B
2x
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C
log2x
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D
logx2
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Solution

The correct option is D log2x
We have, f(xy)=f(x)+f(y)(1)
Put x=y=1
f(1)=0
Now f(x)=limh0f(x+h)f(x)h=limh0f[x(1+h/x)])f(x)h
=limh0f(x)+f(1+h/x)f(x)h=limh0f(1+h/x)f(1)h/x1x=f(1)x
Now integrating we get, f(x)=f(1)logx+c
Since f(1)=0c=0
Thus f(x)=f(1)logx
Also f(2)=11=f(1)log2f(1)=1log2
Hence f(x)=logxlog2=log2x

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