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Question

If P denotes the product of all the values of x satisfying log3x3x+(log3x)2=1, then find 18P

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Solution

log3x3x+(log3x)2=1
Clearly x>0

log33xlog3(3x)+(log3x)2=1[logba=logalogb]

log33logxlog33+log3x+(log3x)2=1

Put t=log3x to obtain

1t1+t+t2=1[logaa=1]

t3+t22t=0
t(t2+t2)=0

t((t1)(t+2))=0

t=0,1,2
x=30,31,32

x=1,3,19
P=19×1×3

P=13

18P=183=6

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