解方程log以(x+2)为底(4x+5)-log以(4x+5)为底(x²+4x+4)-1=0
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解方程log以(x+2)为底(4x+5)-log以(4x+5)为底(x²+4x+4)-1=0
解方程log以(x+2)为底(4x+5)-log以(4x+5)为底(x²+4x+4)-1=0
解方程log以(x+2)为底(4x+5)-log以(4x+5)为底(x²+4x+4)-1=0
log以(4x+5)为底(x²+4x+4)
=log以(4x+5)为底(x+2)²
=2log以(4x+5)为底(x+2)
令log以(x+2)为底(4x+5)=t
则t-2/t-1=0
∴ t²-t-2=0
∴ (t-2)(t+1)=0
即 t=2或t=-1
(1)log以(x+2)为底(4x+5)=2
即 (x+2)²=4x+5
∴ x²+4x+4=4x+5
∴ x=1或x=-1
检验,x=-1不满足,x=1满足
(2)log以(x+2)为底(4x+5)=-1
即 (x+2)^(-1)=4x+5
∴ (4x+5)(x+2)=1
∴ 4x²+13x+9=0
∴ (x+1)(4x+9)=0
∴ x=-1或x=-9/4
检验,x=-1不满足,x=-9/4不满足(底数大于0且不等于1)
综上,方程的根是x=1
log以(x+2)为底(4x+5)-log以(4x+5)为底(x²+4x+4)-1=0
log以(x+2)为底(4x+5)-log以(4x+5)为底(x+2)²-1=0
log以(x+2)为底(4x+5)-2log以(4x+5)为底(x+2)-1=0
log以(x+2)为底(4x+5) - 2 / log以(x+2)为底(4x+5) - 1 = 0
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log以(x+2)为底(4x+5)-log以(4x+5)为底(x²+4x+4)-1=0
log以(x+2)为底(4x+5)-log以(4x+5)为底(x+2)²-1=0
log以(x+2)为底(4x+5)-2log以(4x+5)为底(x+2)-1=0
log以(x+2)为底(4x+5) - 2 / log以(x+2)为底(4x+5) - 1 = 0
设:log以(x+2)为底(4x+5) = t
t-2/t-1=0
t²-t-2=0
t=-1,或者t=2
即:log以(x+2)为底(4x+5) = -1,或者,log以(x+2)为底(4x+5) = 2
即:x+2=1/(4x+5),或,(4x+5)=(x+2)²
解得,x = -1,x = -2.25,x = 1
经检验只有x=1是原方程的解
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