Задание 3. Решить уравнение: 1) 1259* – 120 15* + 12 · 25* = 0; - 2) 144* - 356* + 18 · 9* = 0; 3) 9.4* - 46. 10x – 52x+2 = 0.

Задание 3.1

125 ⋅ 9^x – 120 ⋅ 15^x + 12 ⋅ 25^x = 0

125 ⋅ (3^x)² – 120 ⋅ 3^x ⋅ 5^x + 12 ⋅ (5^x)² = 0

Разделим обе части уравнения на (5^x)²:

125 ⋅ (3^x / 5^x)² – 120 ⋅ (3^x / 5^x) + 12 = 0

Пусть t = (3/5)^x:

125t² - 120t + 12 = 0

D = (-120)² - 4 * 125 * 12 = 14400 - 6000 = 8400

t = (120 ± √8400) / 250 = (120 ± 20√21) / 250 = (12 ± 2√21) / 25

(3/5)^x = (12 ± 2√21) / 25

x = log_3/5((12 ± 2√21) / 25)

Ответ: x = log_3/5((12 + 2√21) / 25), x = log_3/5((12 - 2√21) / 25)

Задание 3.2

14 ⋅ 4^x – 35 ⋅ 6^x + 18 ⋅ 9^x = 0

14 ⋅ (2^x)² – 35 ⋅ 2^x ⋅ 3^x + 18 ⋅ (3^x)² = 0

Разделим обе части уравнения на (3^x)²:

14 ⋅ (2^x / 3^x)² – 35 ⋅ (2^x / 3^x) + 18 = 0

Пусть t = (2/3)^x:

14t² - 35t + 18 = 0

D = (-35)² - 4 * 14 * 18 = 1225 - 1008 = 217

t = (35 ± √217) / 28

(2/3)^x = (35 ± √217) / 28

x = log_2/3((35 ± √217) / 28)

Ответ: x = log_2/3((35 + √217) / 28), x = log_2/3((35 - √217) / 28)

Задание 3.3

9 ⋅ 4^x – 46 ⋅ 10^x – 5^2x+2 = 0

9 ⋅ (2^x)² – 46 ⋅ 2^x ⋅ 5^x – 25 ⋅ (5^x)² = 0

Разделим обе части уравнения на (5^x)²:

9 ⋅ (2^x / 5^x)² – 46 ⋅ (2^x / 5^x) – 25 = 0

Пусть t = (2/5)^x:

9t² - 46t - 25 = 0

D = (-46)² - 4 * 9 * (-25) = 2116 + 900 = 3016

t = (46 ± √3016) / 18 = (23 ± √754) / 9

(2/5)^x = (23 ± √754) / 9

x = log_2/5((23 ± √754) / 9)

Ответ: x = log_2/5((23 + √754) / 9), x = log_2/5((23 - √754) / 9)