Find the last three digits of the product of the positive roots of:
3√(2021)* Xlog2021X = X3
Start by taking log_2021 on both sides. That yields:
(1/3) + log_2021(x)*log_2021(x) = 3*log_2021(x)
For brevity, let L=log_2021(x). Then the last equation can be rewritten as 3L^2 - 9L + 1 = 0.
Then the quadratic formula yields L = (9+sqrt(69))/6 or (9-sqrt(69))/6.
Then x = 2021^(9+sqrt(69))/6 or 2021^(9-sqrt(69))/6
The product of these solutions for x is
[2021^(9+sqrt(69))/6] * [2021^(9-sqrt(69))/6]
= 2021^[(9+sqrt(69))/6 + (9-sqrt(69))/6]
= 2021^3 = 8254655261.
The last three digits of the product of the roots are 261.
Analytic Solution
