Find the product of positive roots of equation sqrt(2008)x^(log_(2008)x)=x^2

Charlie Conner 2022-09-06 Answered
Find the product of positive roots of equation 2008 x log 2008 x = x 2
Problem : Find the product of positive roots of equation 2008 x log 2008 x = x 2
Solution :
The given equation can be written as 2008 x log 2008 x = x 2
2008 x log x 2008 1 = x 2
2008 1 2008 = x 2 [by using a log a m = m]
x 2 = 1 2008
Now how to find the product of positive roots please guide, thanks.
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Answers (1)

odejicahfc
Answered 2022-09-07 Author has 10 answers
Take y = log x log 2008 . Hence
2008 x y = x 2 => 1 2 log 2008 + y log x = 2 log x => 1 2 log x y + y log x = 2 log x => log x ( y + 1 2 y ) = 2 log x => y + 1 2 y = 2 => 2 y 2 4 y + 1 = 0
Therefore
y 1 + y 2 = 2 => l o g ( x 1 x 2 ) = 2 log 2008 => x 1 x 2 = 2008 2
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