Find the last three digits of the product of the positive roots of sqrt(1995)x^(log_(1995)x)=x^2

Sasha Hess

Sasha Hess

Answered question

2022-09-05

Find the last three digits of the product of the positive roots of 1995 x log 1995 x = x 2

Answer & Explanation

Andrejkoxg

Andrejkoxg

Beginner2022-09-06Added 20 answers

1995 x log 1995 x = x 2
applying logarithm on both sides , we get 1 2 log 1995 + log 1995 x log x = 2 log x
1 2 log 1995 = ( 2 log 1995 x ) log x.
log 1995 1 2 log x = 2 log 1995 x
log x 1995 1 2 = + 1 log x 1995 = 2
suppose log x 1995 = a
then this expression becomes 1 2 a + 1 q = 2
a 2 + 2 2 a = 2
a 2 4 a 2 = 0
a = 4 ± 16 + 8 2 = 4 ± 2 6 2 = 2 ± 6
considering a = 2 + 6 , we have
log x 1995 = 2 + 6
x 2 + 6
=1995
x = ( 1995 ) 1 2 + 6 = 5.51626654.

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