# How to Solve an Exponential Equation?

## Solve the exponential equation

Solve the exponential equation **6 ^{x} = 3^{2x} – 2^{2x}**

## 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 = a^{2} – 1

⇒ a^{2} – a – 1 = 0

This is a quadratic equation, so apply 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 logarithm in 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)**