1.
а) 4^(log₂6-0,5) = 2^2(log₂6-0,5) = 2^2log₂6-1=(2^log₂6^2)/2^1 = (2^log₂36)/2 = 36/2=18
б) log₄log₁₄196+log₅√5 = log₄2+1/2 = log₄√4+1/2 = 1/2+1/2 = 1
2.
log₂(2²ˣ +16ˣ) = 2log₄12
log₂(2²ˣ+16ˣ) = log₄¹⁾²12
log₂(2²ˣ+2⁴ˣ) = log₂12
2²ˣ+2⁴ˣ = 12
(2²ˣ)²+2²ˣ-12=0
2²ˣ = a, где а>0
a²+a-12=0
a₁+a₂ = -1
a₁*a₂ = -12
a₁=3
a₂= -4 (<0, не подходит по условию)<br>2²ˣ = 3
ln2²ˣ = ln3
2xln2 = ln3
2x = ln3/ln2
2x= log₂3
x=1/2 log₂3
3.
log₄x²*log₄16/x≥2
2log₄x*(log₄16-log₄x)≥2
2log₄x*(2-log₄x)≥2
log₄x*(2-log₄x)≥1
2log₄x - log₄²x≥1
log₄²x-2log₄x+1≤0
(log₄x-1)²≤0
log₄x-1≤0
log₄x≤1
log₄x≤log₄4
ОДЗ: x>0
x≤4
Ответ: х∈(0; 4]
4.
3ⁿ+x = 10
n - log₃x = 2
x=10-3ⁿ
n = log₃x+ 2
x=10 - 3^(log₃x+2)
n =log₃x+ 2
x= 10-(3^log₃x)*3²
n =log₃x+2
x= 10-9x
n = log₃x+2
10x=10
n = log₃x+2
x = 1
n = log₃1+2
x = 1
n = 2
Ответ: (1;2)