Solving Exponential Equations: A Deep Dive into 5x3 253x - 4

In this article, we will explore a specific type of equation known as an exponential equation, focusing on the problem 5x3 253x - 4. By breaking down the steps and providing detailed solutions, we aim to enhance understanding and proficiency in solving such equations.

Understanding the Problem

The given equation is 5x3 253x - 4. Our goal is to solve for the value of x. Let's begin by simplifying and solving the equation step by step.

Step-by-Step Solution

First, let's rewrite the equation, noting that 25 can be expressed as 52:

5x3 52(3x - 4)

Since the bases are the same, we can set the exponents equal to each other:

x3 2(3x - 4)

This simplifies to:

x3 6x - 8

Next, we rearrange the equation to solve for x:

6x - x 8 3

Simplifying further, we get:

5x 11

Finally, we solve for x:

x 11/5

Therefore, the value of x is 11/5 or 2.2.

Alternative Methods

There are several ways to approach this problem, and we will provide an alternative solution for clarity. Another way to solve the equation is to directly equate the bases and then solve the equation:

5x3 253x - 4

Or, rewriting it as:

5 ^ x3 5 ^ 6x - 8

Setting the exponents equal, we get:

5x - 3 6x - 8

Which simplifies to:

5x - 6x -8 3

Or:

-x -5

Multiplying both sides by -1, we get:

x 5/5 1

Alternatively, if we solve it directly:

5x3 253x - 4

Or, rewriting it as:

5 ^ x3 5 ^ 6x - 8

Setting the exponents equal, we get:

x3 6x - 8

X3 8 6x

5x 11

x 11/5

Key Concepts Recap

The key to solving exponential equations is to ensure the bases are the same. Once the bases are the same, we can set the exponents equal to each other and solve for the variable. This process is often simplified by recognizing common bases and using algebraic manipulation.

Conclusion

In conclusion, we have explored the exponential equation 5x3 253x - 4 and demonstrated that the solution is x 11/5 or 2.2. By following these steps and understanding the underlying principles, you can apply similar methods to solve various exponential equations. This process is crucial for students and professionals alike who work with exponential functions and equations.

Resources and Further Reading

For further practice and detailed explanations, you may find the following resources helpful:

Khan Academy: Exponential Equations Math Is Fun: Exponential Equations Math Planet: Exponential Equations