Tasty Logarithm #5

Algebra Level 4

1995 x log 1995 x = x 2 \large \sqrt{1995}x^{\log_{1995}x}=x^2

Let x 1 , x 2 . . . . x n x_1,x_2....x_n be the positive roots of the equation above.

Find ( x 1 x 2 x 3 x n ) ( m o d 1000 ) (x_1\cdot x_2\cdot x_3\cdot \cdot \cdot x_n) \pmod {1000} .

This is the part of this !!


The answer is 25.

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1 solution

Chew-Seong Cheong
May 13, 2015

1995 x log 1995 x = x 2 log 1995 ( 1995 x log 1995 x ) = log 1995 x 2 log 1995 1995 + log 1995 x log 1995 x = log 1995 x 2 1 2 + log 1995 x ˙ log 1995 x = 2 log 1995 x ( log 1995 x ) 2 2 log 1995 x + 1 2 = 0 \begin{aligned} \sqrt{1995}x^{\log_{1995}{x}} & = x^2 \\ \log_{1995} {\left(\sqrt{1995}x^{\log_{1995}{x}}\right)} & = \log_{1995} {x^2} \\ \log_{1995} {\sqrt{1995}} + \log_{1995} {x^{\log_{1995}{x}}} & = \log_{1995} {x^2} \\ \frac{1}{2} + \log_{1995} {x}\dot{}\log_{1995} {x} & = 2\log_{1995} {x} \\ \Rightarrow \left( \log_{1995} {x} \right)^2 - 2\log_{1995} {x} + \frac{1}{2} & = 0 \end{aligned}

There are two roots x 1 x_1 and x 2 x_2 , and by Vieta's formulas, we have the sum of roots:

log 1995 x 1 + log 1995 x 2 = log 1995 x 1 x 2 = 2 x 1 x 2 = 199 5 2 = 3980025 ( x 1 x 2 ) m o d 1000 = 25 \log_{1995} {x_1} + \log_{1995} {x_2} = \log_{1995} {x_1x_2} = 2 \\ \Rightarrow x_1x_2 = 1995^2 = 3980025\\ \Rightarrow \left(x_1x_2\right) \mod{1000} = \boxed{25}

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