## 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 = -13/2*x*:

### Solution

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