1. (log0,2_x)^2 ≤ 1; x > 0
- 1 ≤ log1/5_ x ≤ 1;
log1/5_5 ≤ log1/5_ x ≤ log1/5_1/5;
1/5 < 1; ⇒ 1/5 ≤ x ≤ 5.
x ∈ [0,2; 5].
3. (1/4)^x - 2^(1-x) - 8 < 0;
(2^-x)^2 - 2* 2^-x - 8 < 0;
2^-x = t > 0;
t^2 - 2t - 8 =0;
D = 4+32 = 36 = 6^2;
t1 = (2+6) / 2 = 4; t2 = (2-6) / 2 = - 2 < 0
2^-x = 4;
2^-x = 2^2;
-x = 2;
x = -2.
4. (1/2)^x + (1/2)^(x - 2) > 5;
(1/2)^x + 1/2^(-2) *(1/2)^x > 5;
(1/2)^x + 4*(1/2)^x > 5;
5*(1/2)^x > 5;
(1/2)^x > 1;
(1/2)^x > (1/2)^0;
1/2 < 1; ⇒ x < 0.
6. {2x - 1 > 0; {2x > 1; { x > 0,5;
3x - 4 > 0; 3x > 4/; x > 4/3;
2x - 1 > 3x - 4; - x > - 3; x < 3. ⇒ x ∈(4/3; 3).
8. (2*lgx)^2 + 3 * lg x > 1;
lg x = t; х > 0
4 t^2 + 3 t > 1;
4t^2 + 3t - 1 > 0;
4(t-1)(t+1/4) > 0;
+ - +
___(-1/4)____(1)_____ t
t < - 1/4 U t > 1
lg x < - 1/4 U lg x > 1;
x < 1/ корень 4-й степени из 10 или x > 10.
Ответ
х ∈(0; 1/ корень4-й степени из 10) ∨ ( 10; + бесконечность).
8. 7^2x - 3*7^x > 10;
7^x = t > 0;
t^2 - 3t - 10 > 0;
(t-5)(t+2) > 0;
+ - +
____(-2)___(5)____t
t > o; ⇒ t > 5;
7^x > 5;
x > log5_7