Problem:
Find the last three digits of the product of the positive roots of
1995xlog1995x=x2
Solution:
Taking logarithms of both sides of the equation, we find that
log(1995xlogx)=logx2
where all the logarithms are to the base 1995. From this we obtain
21log1995+(logx)(logx)=2logx
which leads to
(logx)2−2logx+21=0
Solving this last equation gives
logx=1±21
Since these values for logx are both real, the original equation has two positive roots; call them r1 and r2. Since logr1r2=logr1+logr2=2, the product of these roots is