5^(x+2) + 2*5^x ≤ 51
25* 5^x + 2/5^x ≤ 51
5^x=t t>0
25t + 2/t -51 ≤ 0
25t²-51*t + 2 ≤ 0
D=2601-200=2401=49²
t12=(51+-49)/50 = 2 1/25
t = [1/25 2]
5^x≥5^-2 x≥-2
5^x≤2 x≤ log(5) 2
x⊂[-2 log(5) 2] log(5) 2 ≠0.43
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log(2x) 0.25 ≥ log(2) 32x - 1
ОДЗ x>0 x≠1/2
log(2x) 0.25= log(2x) 1/4 = log(2x) 2^-2 = -2 log(2x) 2 = -2/ log(2) 2x
log(2) 32x - 1 = log(2) 2⁴*2x - 1= log(2) 2x + 4 -1=log(2) 2x + 3
-2/ log(2) 2x ≥ log(2) 2x + 3
log(2) 2x = t
t + 3 + 2/t ≤ 0
(t²+3t+2)/t = (t+1)(t+2)/t ≤ 0
////////-///// [-2] /////// + /////// [-1] ///////-/////// (0) /////////+///////
t=(-∞ -2] U [-1 0)
log(2) 2x ≤-2
2x≤1/4
x≤1/8 = 0.125
log(2) 2x ≥ -1
2x≥1/2
x≥1/4 = 0.25
log(2) 2x <0<br>2x<1<br>x<1/2<br>========================
x⊂[-2 log(5) 2] log(5) 2 ≠0.43
x>0 x≠1/2
x≤1/8 = 0.125
1/2>x≥1/4 = 0.25
ответ x=(0 1/8] U [1/4 log(5) 2]