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Question

If x{34(log3x)2+(log3x)54}=3, then x has

A
all integral values
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B
two integral values and one irrational value
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C
all irrational values
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D
two rational values and one irrational value
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Solution

The correct option is D two rational values and one irrational value
Clearly, x>0
x{34(log3x)2+(log3x)54}=3
34(log3x)2+(log3x)54=logx3
34(log3x)2+(log3x)54=12log3x
3(log3x)3+4(log3x)25(log3x)2=0, log3x0

Let log3x=t
Then 3t3+4t25t2=0, t0
Clearly t=1 is a solution.
So, (t1)(3t2+7t+2)=0
(t1)(t+2)(3t+1)=0
t=1,2,13
log3x=1,2,13
x=3,19,133

Clearly, x takes two rational values and one irrational value.

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