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Suppose that a and b are positive real n...

Suppose that a and b are positive real numbers such that `log_(27)a+log_9(b)=7/2` and `log_(27)b+log_9a=2/3`.Then the value of the `ab` equals

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`log_(27)a+log_(9)b= 7/2 and log_(27)b+log_(9) a = 2/3`
` rArr 1/3 log _(3) a + 1/2 log _(3) b= 7/2`
` and 1/3 log_(3) b+1/2 log _(3) a = 2/3`
Adding the equations, we get
`1/3 log_(3)(ab)+1/2 log_(3)(ab) = 7/2+2/3 = 25/6`
` or 5/6 log_(3) (ab) = 25/6`
`or log_(3) (ab) = 5 `
` or ab = 3^(5) = 243`
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