Question

2 pow(x-1)=3 power y and 4 pow x =3 pow(y+1)).find x and y

Answer 1

The required solution is

x = log₂(3/2), y = log₃(3/4)

Step-by-step explanation:

Given,

2ˣ⁻¹ = 3ʸ ..... (1)

or, 2ˣ * 2⁻¹ = 3ʸ

and 4ˣ = 3ʸ⁺¹

or, (2²)ˣ = 3ʸ * 3¹

or, 2²ˣ = 2ˣ⁻¹ * 3 [by (1)]

or, 2²ˣ / 2ˣ⁻¹ = 3

or, 2²ˣ⁻ˣ⁺¹ = 3

or, 2ˣ⁺¹ = 3

or, 2ˣ * 2¹ = 3

or, 2ˣ * 2 = 3

or, 2ˣ = 3/2 ..... (2)

Taking log in both sides, we get

log(2ˣ) = log(3/2)

or, x log2 = log(3/2)

or, x = log₂(3/2)

From (1), we get

3ʸ = 2ˣ⁻¹

or, 3ʸ = 2ˣ * 2⁻¹

or, 3ʸ = 3/2 * 1/2 [by (2)]

or, 3ʸ = 3/4

Taking log in both sides, we get

log(3ʸ) = log(3/4)

or, y log3 = log(3/4)

or, y = log(3/4) / log3

or, y = log₃(3/4)

Therefore the required solution is

x = log₂(3/2), y = log₃(3/4)