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Question

If S be a set defined as
S={xR:log5120+(x3)2log5(15x3)=log5(0.25x4)}, then which of the following statement is true?

A
S is an empty set
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B
S contains exactly one element
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C
S contains exactly two elements
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D
S contains exactly three elements
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Solution

The correct option is B S contains exactly one element
For the given equation to be defined, 15x3>0 and 0.25x4>0
5x3<1 and 5x4<0.2
5x3<1 and 5x4<15
x<3 and x<3
x<3(1)

log5120+(x3)2log5(15x3)=log5(0.25x4)
log5120+log55x3log5(15x3)2+log5(0.25x4)=0
log5120×5x3(0.25x4)(15x3)2=0
120×5x3(0.25x4)(15x3)2=1
120×5x3(15x3)(15x3)2×5=1
24×5x3(15x3)(15x3)2=1

Assuming 5x3=t, we get
24×t(1t)(1t)2=124t(1t)=(1t)2(1t)[24t(1t)]=0t=1 or t=1255x3=1 or 5x3=125x3=0 or x3=2x=3 or x=1
As x<3x=1
S={1}

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