1)
lg300-lg3+100^lg6=lg(3*100)-lg3+10^(2*lg6)=
=lg3+lg10²-lg3+10^lg36=2+36=38.
2) a)
log₄(x+1)+log₄(x+1)²=3
log₄(x+1)³=3
(x+1)³=4³
x+1=4
x=3.
б)
lg²x-5=4*lg(1/x)
lg²x-4lgx⁻¹-5=0
lg²x+4lgx-5=0
lgx=t
t²+4t-5=0 D=36
t₁=1 lgx=1 x₁=10¹=10
t₂=-5 lgx=-5 x₂=10⁻⁵
3)
log₂/₃(2-5x)<-2 ОДЗ: 2-5x>0 x<2,5<br>2-5x>(2/3)⁻²
2-5x>(3/2)²
2-5x>2,25
5x<-0,25<br>x<-0,05<br>4)
y=x*e^(-x)
y`=(x*e^(-x))`=0
e^(-x)+(x*(-e^(-x))=e^(-x)*(1-x)=0
e^(-x)>0 ⇒
1-x=0
x=1
y=x*e^(-x)=1*e^(-1)=1/e≈0,368
Ответ: (1;1/e).
5)
log₂(x+y)+2*log₄(x-y)=5 log₂(x+y)+2*log₂²(x-y)=log₂32
3^(1+2*log₃(x-y)=48 3*3^log₃(x-y)²=48 I÷3
ОДЗ: x+y>0 x-y>0
log₂(x+y)+2*(1/2)*log₂(x-y)-log₂32=0
3^log₃(x-y)²=16
log₂(x+y)+log₂(x-y)-log₂32=0 log₂(x²-y²)/32=0 (x²-y²)/32=1
(x-y)²=4² x-y=4
x²-y²=32 (x-y)(x+y)=32 4*(x+y)=32 x+y=8
x-y=4 x-y=4 ⇒
2x=12
x=6 y=2.