Показательные уравнения:

1. (1/3) ^2x = 162 - (1/3) ^x-2

2. (1/2) ^2x = 32 + (1/2) ^x-2

1. (1/3) ²ˣ = 162 - (1/3) ˣ-2

Замена переменной:

(1/3) ˣ=t; t>0

t² + (1/3) ⁻²·t - 162=0

t²+9t-162=0

D=81+4·162=81· (1+8) = 81·9 = (9·3) ²=27²

t = (-9-27) / 2=-18 0 или

t = (-9+27) / 2=9

(1/3) ˣ=9

x=-2

О т в е т. - 2.

2. (1/2) ²ˣ = 32 + (1/2) ˣ-2

Замена переменной:

(1/2) ˣ=t; t>0

t² - (1/2) ⁻²·t - 32=0

t²-4t-32=0

D=16+4·32=16· (1+8) = 16·9 = (4·3) ²=12²

t = (4-12) / 2=-4 0 или

t = (4+12) / 2=8

(1/2) ˣ=8

x=-3

О т в е т. - 3.