令
X=lg(2)
Y=lg(5)
lg^3(2) + lg^3(5))+ 3lg(2)lg(5)
= X^3 + Y^3 + 3XY
= (X+Y) (X^2 -XY + Y^2) + 3XY
一.X+Y = log(2) + log(5) = log(2*5) = log(10) = 1
所以,
lg^3(2) + lg^3(5))+ 3lg(2)lg(5)
= (X+Y) (X^2 -XY + Y^2) + 3XY
= 1 * (X^2 -XY + Y^2) + 3XY
= X^2 +2XY + Y^2
= (X+Y)^2
= 1^2
= 1