Verified answer

log(2) (4^x + 4) = x + log(2) (2^x*2^1 - 3)

log(2) (4^x + 4) = x + log(2) (2^(x+1) - 3)

ОДЗ

4^x + 4 > 0  x∈ R

2^(x+1) >  3

log(2) 2^(x+1) > log(2) 3

x + 1 > log(2) 3

x > log(2) 3 - 1  ≈ 1.59 - 1 ≈ 0.59

ОДЗ x ∈ (log(2) 3 - 1 , +∞ )

log(2) (4^x + 4) = x + log(2) (2^(x+1) - 3)

log(2) (4^x + 4) = log (2) 2^x + log(2) (2^(x+1) - 3)

log(2) (4^x + 4) = log(2) 2^x*(2*2^x - 3)

снимаем логарифмы

4^x + 4 = 2^x*(2*2^x - 3)

(2^x)^2 + 4 = 2*2^x*2^x - 3*2^x

(2^x)^2 - 3*2^x - 4 = 0

2^x = t > 0

t^2 - 3t - 4 = 0

D=9 + 16 = 25 = 5²

t₁₂ = (3 +- 5)/2 = -1   4

1. t₁ = -1

решений нет t>0

2. t=4

2^x = 4

x = 2 (входит в ОДЗ x > log(2) 3 - 1 )

ответ х=2