Question

计算2lg5+2/3lg8+lg5·lg20+(lg∧2)2的值. 解方程以4为底3-X的对数+以0.25为底3十X

计算2lg5+2/3lg8+lg5·lg20+(lg∧2)2的值.
解方程以4为底3-X的对数+以0.25为底3十X的对数=以4底1-X的对数十以0.25为底2X十1的对数.

Answer 1

计算2lg5+2/3lg8+lg5·lg20+(lg∧2)2的值.
2lg5+2/3lg8+lg5·lg20+(lg∧2)2
=2lg5+2lg8^1/3+lg5*lg(5*4)+(lg2)²
=2lg5+2lg2+lg5(lg5+2lg2)+(lg2)²
=2lg5+2lg2+(lg5)²+2lg5*lg2+(lg2)²
=2(lg5+lg2)+(lg5+lg2)²
=(lg2+lg5+1)²-1
解方程以4为底3-X的对数+以0.25为底3十X的对数=以4底1-X的对数十以0.25为底2X十1的对数.
log4 (3-x)+log0.25 (3+x)=log4 (1-x)+log0.25 (2x+1)
log4 (3-x)+log4 (3+x)/(log4 0.25)=log4 (1-x)+log4 (2x+1)/(log4 0.25)
log4 (3-x)-log4 (3+x)=log4 (1-x)-log4 (2x+1)
log4 (3-x)/(3+x)=log4 (1-x)/(2x+1)
(3-x)/(3+x)=(1-x)/(2x+1)
(3-x)(2x+1)=(1-x)(3+x)
-2x²+5x+3=-x²-2x+3
x²-7x=0
x(x-7)=0
x1=0
x2=7

Answer 2

2lg5+2/3lg8+lg5lg20+(lg2)^2
=2lg5+2lg2+lg5(1+lg2)+(lg2)^2
=2(1-lg2)+2lg2+(1-lg2)(1+lg2)+(lg2)^2
=2-2lg2+2lg2+1-(lg2)^2+(lg2)^2
=3