log以3为底x的对数的号外平方+log以9为底3x的对数=2

Question

如题所述

Answer 1

[log3(x)]²+log9(3x)=2

log9(3x)=(1/2)log3(3x)=(1/2)[log3(3)+log3(x)]=1/2+(1/2)log3(x)

所以，原方程化为：
[log3(x)]²+(1/2)log3(x)+1/2=2
即：[log3(x)]²+(1/2)log3(x)-3/2=0
2[log3(x)]²+log3(x)-3=0
令log3(x)=t，则2t²+t-3=0
(t-1)(2t+3)=0
t1=1，t2=-3/2
log3(x)=1，得：x1=3
log3(x)=-3/2，得：x2=3^(-3/2)=(√3)/9
所以，原方程的解为：x1=3，x2=(√3)/9

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Answer 2

(log3 x)^2+log9 (3x)=2

(lgx/lg3)^2+(lg3x)/lg9=2
(lgx)^2/(lg3)^2+(lg3+lgx)/2lg3=2
2(lgx)^2+(lg3+lgx)lg3=4(lg3)^2
2(lgx)^2+lg3lgx-3(lg3)^2=0
(2lgx+3lg3)(lgx-lg3)=0
2lgx+3lg3=0, lgx^2=lg3^(-3), x^2=3^(-3), x=根号3/9
lgx-lg3=0, lgx=lg3, x=3

Answer 3

[log3(x)]^2+log9(3x)=2
令log3(x)=t
log9(3x)=(1/2)[1+log3(x)]=1/2+t/2
t^2+t/2+1/2=2
2t^2+t-3=0
(2t+3)(t-1)=0
t=-3/2或t=1
x=3^(-3/2)或x=3

Answer 4

X=根号3