How to Solve the Exponential Equation?

Solve the exponential equation

Solve the exponential equation 6^x = 3^2x – 2^2x

The solution to the exponential equation

6^x = 3^2x – 2^2x

⇒ 3^x 2^x = 3^x 3^x – 2^x 2^x

⇒ 1 = (3^x 3^x – 2^x 2^x)/(3^x 2^x)

⇒ 1 = 3^x/2^x – 2^x/3^x

⇒ 1 = (3/2)^x – (2/3)^x

Let a = (3/2)^x then 1/a = (2/3)^x

⇒ 1 = a – 1/a

⇒ a = a² – 1

⇒ a² – a – 1 = 0

This is a quadratic equation, so apply the quadratic formula

⇒ a = (1 ± √(1 – 4 × 1 × (-1)))/(2 × 1)

⇒ a = (1 ± √5)/2

⇒ a = (1 + √5)/2 (because (3/2)^x is a real value)

Take logarithms on both sides

log (3/2)^x = log ((1 + √5)/2)

⇒ x log (3/2) = log (1 + √5) – log 2

⇒ x = (log (1 + √5) – log 2)/(log 3 – log 2)