Solving Exponential Equations A Step-by-Step Guide

Exponential equations, where the variable appears in the exponent, are a fundamental topic in mathematics. Mastering the techniques to solve these equations is crucial for various fields, including physics, engineering, and finance. In this article, we will delve into a step-by-step approach to solve the elementary exponential equation:

$(\frac{2}{3})^{5x+2} = (\frac{4}{9})^3$

This comprehensive guide will not only provide the solution but also equip you with the underlying principles and strategies to tackle similar problems with confidence.

Understanding Exponential Equations

Before diving into the solution, let's establish a clear understanding of exponential equations. An exponential equation is an equation in which the variable occurs in the exponent. These equations often involve exponential functions, where a constant base is raised to a variable power. The key to solving exponential equations lies in manipulating the equation to isolate the variable. This often involves using properties of exponents and logarithms.

Exponential equations are a cornerstone of mathematical analysis and have extensive applications across diverse fields. From modeling population growth in biology to calculating compound interest in finance, understanding and solving exponential equations is essential. To effectively tackle these equations, a solid grasp of exponent rules and logarithmic properties is paramount. This article aims to provide a comprehensive understanding of solving exponential equations, specifically focusing on an elementary example. By breaking down the steps and explaining the underlying principles, we will equip you with the knowledge and skills to confidently approach a wide range of exponential equation problems.

The Importance of Exponent Rules

The rules of exponents are the foundation for simplifying and manipulating exponential expressions. Here are some key rules that we will utilize in solving the equation:

• Product of Powers: $a^m * a^n = a^{m+n}$
• Quotient of Powers: $a^m / a^n = a^{m-n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Power of a Product: $(ab)^n = a^n * b^n$
• Power of a Quotient: $(a/b)^n = a^n / b^n$
• Negative Exponent: $a^{-n} = 1/a^n$
• Zero Exponent: $a^0 = 1$ (where a ≠ 0)

These rules allow us to rewrite exponential expressions in different forms, making it easier to compare and solve equations. In the context of our given equation, we will primarily focus on manipulating the bases to be the same, which will then allow us to equate the exponents.

Solving the Equation: A Step-by-Step Approach

Now, let's tackle the equation step-by-step:

$(\frac{2}{3})^{5x+2} = (\frac{4}{9})^3$

Step 1: Express Both Sides with the Same Base

The first crucial step in solving this exponential equation is to express both sides of the equation with the same base. This allows us to directly compare the exponents. Observe that $\frac{4}{9}$ can be written as $(\frac{2}{3})^2$. Therefore, we can rewrite the equation as:

$(\frac{2}{3})^{5x+2} = ((\frac{2}{3})^2)^3$

This transformation is the key to unlocking the solution. By recognizing the relationship between $\frac{4}{9}$ and $\frac{2}{3}$, we can create a common base, which is essential for the next steps. This step highlights the importance of recognizing patterns and equivalent forms of numbers in mathematical problem-solving. The ability to identify common bases is a valuable skill that can be applied to a wide variety of exponential equations.

Step 2: Simplify the Exponent on the Right Side

Using the power of a power rule, which states that $(a^m)^n = a^{mn}$, we can simplify the exponent on the right side of the equation:

$(\frac{2}{3})^{5x+2} = (\frac{2}{3})^{2*3}$

$(\frac{2}{3})^{5x+2} = (\frac{2}{3})^6$

This simplification is a direct application of the exponent rules. By understanding and applying these rules correctly, we can reduce complex expressions to simpler forms. In this case, the power of a power rule allows us to combine the exponents, making the equation easier to manage. This step demonstrates the elegance and efficiency of mathematical rules in simplifying complex problems. The result of this simplification brings us one step closer to isolating the variable and finding the solution.

Step 3: Equate the Exponents

Now that both sides of the equation have the same base, we can equate the exponents. This is a fundamental property of exponential equations: if $a^m = a^n$, then $m = n$. Applying this property, we get:

$5x + 2 = 6$

This step is a critical turning point in solving exponential equations. By equating the exponents, we transform the exponential equation into a simple linear equation. This transformation allows us to use familiar algebraic techniques to isolate the variable. The ability to make this transition is a key skill in solving exponential equations and highlights the interconnectedness of different mathematical concepts. The resulting linear equation is now readily solvable, leading us to the final steps of the solution.

Step 4: Solve for x

We now have a simple linear equation to solve for x. Subtract 2 from both sides:

$5x = 6 - 2$

$5x = 4$

Then, divide both sides by 5:

$x = \frac{4}{5}$

This step involves standard algebraic manipulations to isolate the variable. Subtracting and dividing both sides of the equation maintains the equality while bringing us closer to the solution. These operations are fundamental to solving linear equations and are essential tools in any mathematician's toolkit. The final result, $x = \frac{4}{5}$, is the solution to the exponential equation.

Final Solution

Therefore, the solution to the elementary exponential equation

$(\frac{2}{3})^{5x+2} = (\frac{4}{9})^3$

is:

$x = \frac{4}{5}$

This solution satisfies the original equation, meaning that when $x$ is substituted with $\frac{4}{5}$, both sides of the equation will be equal. Verifying the solution is an important step in problem-solving, as it ensures that the answer is correct and that no errors were made during the process. The solution highlights the power of mathematical techniques in solving complex problems and provides a sense of accomplishment in mastering the concepts.

Verification (Optional)

To verify our solution, we can substitute $x = \frac{4}{5}$ back into the original equation:

$(\frac{2}{3})^{5(\frac{4}{5})+2} = (\frac{4}{9})^3$

$(\frac{2}{3})^{4+2} = (\frac{4}{9})^3$

$(\frac{2}{3})^6 = (\frac{4}{9})^3$

$(\frac{2}{3})^6 = (\frac{2^2}{3^2})^3$

$(\frac{2}{3})^6 = (\frac{2}{3})^{2*3}$

$(\frac{2}{3})^6 = (\frac{2}{3})^6$

Since both sides are equal, our solution is correct.

Conclusion

In this article, we have demonstrated a step-by-step method for solving an elementary exponential equation. The key steps include expressing both sides with the same base, simplifying exponents, equating exponents, and solving the resulting linear equation. By mastering these techniques and understanding the underlying principles, you will be well-equipped to solve a wide range of exponential equations. Remember to always verify your solution to ensure accuracy.

Solving exponential equations is not just a mathematical exercise; it's a gateway to understanding complex phenomena in the real world. From radioactive decay to population growth, exponential functions are used to model a variety of processes. By developing your skills in solving these equations, you are not only enhancing your mathematical abilities but also gaining valuable tools for analyzing and interpreting the world around you. The journey of learning mathematics is a continuous one, and each solved problem brings you one step closer to a deeper understanding of the subject.