(2/3)^5x+1 = (27/8)^x-4

2 min read Jun 16, 2024

Solving Exponential Equations: (2/3)^(5x+1) = (27/8)^(x-4)

This article will guide you through the process of solving the exponential equation (2/3)^(5x+1) = (27/8)^(x-4). We will utilize the properties of exponents and logarithms to find the solution for x.

Understanding the Problem

Our goal is to isolate x in the given equation. To do this, we need to express both sides of the equation with the same base. This will allow us to equate the exponents and solve for x.

Finding a Common Base

Simplify (27/8): Notice that (27/8) can be expressed as (3/2)³.
Rewriting the equation: Now, we can rewrite the original equation as: (2/3)^(5x+1) = [(3/2)³]^(x-4)
Using exponent rules: Applying the rule (a^m)^n = a^(m*n), we get: (2/3)^(5x+1) = (3/2)^(3x-12)

Equating Exponents

Since we have the same base on both sides, we can equate the exponents: 5x + 1 = 3x - 12

Solving for x

Now, we have a simple linear equation. Let's solve for x:

Combine like terms: 2x = -13
Isolate x: x = -13/2

Solution

Therefore, the solution to the equation (2/3)^(5x+1) = (27/8)^(x-4) is x = -13/2.