Question

FIND THE PRODUCT OF THE POSITIVE ROOTS OF THE EQUATION √2009 ( x ) ^log₂₀₀₉x = x²

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Answer 1

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√2009 ( x )^log₂₀₀₉x = x²

Taking log with base 2009 on both the sides

log₂₀₀₉√2009 + log₂₀₀₉ x ( log₂₀₀₉ x ) = log₂₀₀₉ x²

1/2 + (log₂₀₀₉)² = 2 log₂₀₀₉ x

(log₂₀₀₉ x )² -2 log₂₀₀₉ x + 1/2 = 0

And let m and n be the roots ,

log₂₀₀₉m + log₂₀₀₉ n = 2

log₂₀₀₉ ( mn ) = 2

mn = 2009²√2009 ( x )^log₂₀₀₉x = x²

Taking log with base 2009 on both the sides

log₂₀₀₉√2009 + log₂₀₀₉ x ( log₂₀₀₉ x ) = log₂₀₀₉ x²

1/2 + (log₂₀₀₉)² = 2 log₂₀₀₉ x

(log₂₀₀₉ x )² -2 log₂₀₀₉ x + 1/2 = 0

And let m and n be the roots ,

log₂₀₀₉m + log₂₀₀₉ n = 2

log₂₀₀₉ ( mn ) = 2

mn = 2009²

As, (m,n) are roots. m* n is the product of roots.

Hence, Required answer is 2009²