(8^x-2^x)/(6^x-3^x)=2

3 min read Jun 16, 2024

Solving the Equation (8^x - 2^x) / (6^x - 3^x) = 2

This equation presents a unique challenge due to the presence of multiple exponential terms with different bases. Here's how we can approach solving it:

1. Simplifying the Equation

Rewrite the terms with common bases:
- 8^x can be expressed as (2^3)^x = 2^(3x)
- 6^x can be expressed as (2^1 * 3^1)^x = 2^x * 3^x
Substitute these expressions back into the original equation: (2^(3x) - 2^x) / (2^x * 3^x - 3^x) = 2

2. Factoring and Solving

Factor out common terms:
- Numerator: 2^x(2^(2x) - 1)
- Denominator: 3^x(2^x - 1)
Rewrite the equation: (2^x(2^(2x) - 1)) / (3^x(2^x - 1)) = 2
Simplify by canceling common terms: (2^(2x) - 1) / (3^x(2^x - 1)) = 2

3. Isolating the Variable

Multiply both sides by the denominator: 2^(2x) - 1 = 2 * 3^x(2^x - 1)
Expand the right side: 2^(2x) - 1 = 2 * 2^x * 3^x - 2 * 3^x
Move all terms to one side: 2^(2x) - 2 * 2^x * 3^x + 2 * 3^x - 1 = 0

4. Solving the Equation

Unfortunately, the equation we've arrived at doesn't have a simple algebraic solution. We can explore a few options:

Numerical Methods: Using numerical methods like Newton-Raphson iteration or graphing calculators, we can find approximate solutions for the value of 'x'.
Graphical Approach: By plotting the left-hand side and the right-hand side of the equation separately, we can visually identify the intersection points which represent the solutions.

Conclusion

The equation (8^x - 2^x) / (6^x - 3^x) = 2 doesn't have a straightforward analytical solution. We can utilize numerical or graphical methods to find approximate solutions for the value of 'x'.