(1/3)^x=27^x+2

2 min read Jun 16, 2024

Solving the Exponential Equation: (1/3)^x = 27^(x+2)

This article will guide you through solving the exponential equation (1/3)^x = 27^(x+2). We will break down the steps and utilize key properties of exponents to find the solution.

Understanding the Equation

The equation presents us with two terms involving the variable x in the exponents. To solve this, we need to express both sides of the equation with the same base.

Expressing with the Same Base

Left Side: (1/3)^x can be rewritten as 3^(-x) since (a/b)^n = a^n / b^n
Right Side: 27 is 3 cubed (3^3), so 27^(x+2) can be rewritten as (3^3)^(x+2). Applying the power of a power rule [(a^m)^n = a^(m*n)], this simplifies to 3^(3x+6).

Solving for x

Now we have the equation 3^(-x) = 3^(3x+6). Since the bases are the same, we can equate the exponents:

-x = 3x + 6

Solving for x:

-4x = 6 x = -6/4 x = -3/2

Solution

Therefore, the solution to the equation (1/3)^x = 27^(x+2) is x = -3/2.