Найти больший корень уравнения (или корень, если он единственный) 5*3^{2x}+15*5^{2x-1}=8*15^{x}

Упростим уравнение:

5 ⋅ 3²ˣ + 15 ⋅ 5²ˣ⁻¹ = 8 ⋅ 15ˣ

5 ⋅ 3²ˣ + 3 ⋅ 5²ˣ = (3 + 5) ⋅ 3ˣ ⋅ 5ˣ

5 ⋅ 3²ˣ + 3 ⋅ 5²ˣ = 3 ⋅ 3ˣ ⋅ 5ˣ + 5 ⋅ 3ˣ ⋅ 5ˣ

5 ⋅ 3ˣ ⋅ (3ˣ - 5ˣ) - 3 ⋅ 5ˣ ⋅ (3ˣ - 5ˣ) = 0

(3ˣ - 5ˣ) ⋅ (5 ⋅ 3ˣ - 3 ⋅ 5ˣ) = 0

Получим совокупность двух уравнений:

3ˣ - 5ˣ = 0

5 ⋅ 3ˣ - 3 ⋅ 5ˣ = 0

Решим первое уравнение:

3ˣ = 5ˣ

x = 0

Решим второе уравнение:

5 ⋅ 3ˣ = 3 ⋅ 5ˣ

(3 / 5) ˣ = 3 / 5

x = 1

Наибольший корень уравнения: x = 1.