4.
a) logₓ1000=3
x³=1000
x³=10³
x=10.
b) log₅(115-x)=1
115-x=5¹
115-x=5
x=110.
c) log₄(x²-4x+1)-log⁴(x²-6x+5)=0
log₄((x²-4x+1)/(x²-6x+5))=0
(x²-4x+1)/(x²-6x+5)=4⁰
(x²-4x+1)/(x²-6x+5)-1=0
(x²-4x+1-x²+6x-5)/(x²-6x+5)=0
(2x-4)/(x²-5x-x+5)=0
2*(x-2)/((x(x-5)-(x-5))=0 |÷2
(x-2)/((x-5)(x-1))=0
x-5≠0 x≠5 x-1≠0 x≠1 ⇒
x-2=0
x=2
Ответ: x=2.
5.
a) log₅(9-x)<2 ОДЗ: 9-x>0 x<9<br>9-x<5²<br>9-x<25<br>x>-16 ⇒
x∈(-16;9).
b) log₀,₃(7-x)≥log₀,₃(8+2x)
ОДЗ: 7-x>0 x<7 8+2x>0 2x>-8 x>-4 x∈(-4;7)
7-x≤8+2x
3x≥-1
x≥-1/3 ⇒
x∈[-1/3;7).