Solving Exponential Equations: Find X In 3^(2x) = 9^(3x-4)

Have you ever been faced with an exponential equation that seemed like a puzzle? Well, you’re in the right place! Exponential equations might look intimidating at first, but with the right approach, they can be quite straightforward to solve. In this article, we'll dive into a specific problem: finding the value of x in the equation 3^(2x) = 9^(3x-4). We'll break down the steps, explain the concepts, and by the end, you'll feel confident tackling similar problems. Understanding exponential equations is crucial in various fields, from finance to science, so let's get started and unravel this mathematical mystery together.

Understanding Exponential Equations

Before we jump into solving our specific equation, let's take a moment to understand what exponential equations are and why they're important. Exponential equations are equations where the variable appears in the exponent. They are used to model many real-world phenomena, such as population growth, radioactive decay, and compound interest. The key characteristic of exponential functions is that they show rapid growth or decay, making them essential in understanding various natural and financial processes. For instance, in finance, understanding exponential growth is crucial for calculating returns on investments over time. In biology, exponential decay helps in determining the half-life of radioactive substances. Mastering exponential equations not only enhances your mathematical skills but also provides a powerful tool for analyzing and predicting real-world trends.

The basic form of an exponential equation is a^x = b, where a is the base, x is the exponent, and b is the result. Solving these equations often involves manipulating the bases to be the same or using logarithms, which are the inverse operation of exponentiation. For our equation, 3^(2x) = 9^(3x-4), the base on the left side is 3, and on the right side, it's 9. Our goal is to find a way to make these bases the same so that we can equate the exponents. This is a common strategy in solving exponential equations and simplifies the process significantly. By understanding the properties of exponents and how to manipulate them, we can transform complex-looking equations into simpler, solvable forms. This foundational knowledge is what we’ll build upon as we solve our problem step by step.

Key Concepts for Solving Exponential Equations

To effectively solve exponential equations, there are a few key concepts we need to keep in mind. First and foremost is the property of equality for exponential functions. This property states that if a^m = a^n, then m = n, provided that a is a positive number not equal to 1. This is the cornerstone of our approach, as it allows us to equate exponents once we have the same base on both sides of the equation. Another crucial concept is the power of a power rule, which states that (a^m)n = a^(m*n). This rule is essential for simplifying expressions where an exponent is raised to another exponent, as we will see in our example. Additionally, understanding how to rewrite numbers as powers of a common base is vital. In our case, recognizing that 9 is a power of 3 (9 = 3^2) is the key to simplifying the equation.

Another important aspect is the ability to manipulate exponents using various exponent rules. For example, the rule a^(m+n) = a^m * a^n can be used in reverse to combine terms with the same base. The rule a^(m-n) = a^m / a^n is also helpful in simplifying exponential expressions. Familiarity with these rules and the ability to apply them flexibly will significantly aid in solving exponential equations. Moreover, it's worth noting that logarithms provide an alternative method for solving exponential equations, especially when it's not straightforward to find a common base. However, for our specific problem, we can solve it elegantly by manipulating the bases and exponents directly, making it a good illustration of the core principles involved in solving exponential equations.

Step-by-Step Solution of 3^(2x) = 9^(3x-4)

Now, let's tackle the equation 3^(2x) = 9^(3x-4) step by step. Our primary goal is to express both sides of the equation with the same base. Currently, we have a base of 3 on the left and a base of 9 on the right. Recognizing that 9 is a power of 3 is the first crucial step. We can rewrite 9 as 3^2. This substitution will allow us to have the same base on both sides, making it easier to compare the exponents.

Step 1: Rewrite 9 as 3^2

So, our equation becomes:

3^(2x) = (3²⁾(3x-4)

Now that we have expressed 9 as a power of 3, we can move on to the next step, which involves simplifying the right side of the equation. This is where the power of a power rule comes into play. This rule will help us eliminate the parentheses and combine the exponents on the right side.

Step 2: Apply the power of a power rule

Using the rule (a^m)n = a^(m*n), we multiply the exponents on the right side:

3^(2x) = 3^(2 * (3x-4))

Now, distribute the 2 in the exponent on the right side:

3^(2x) = 3^(6x-8)

We've successfully rewritten the equation with the same base on both sides. Now we can proceed to equating the exponents, which is a direct application of the property of equality for exponential functions.

Step 3: Equate the exponents

Since the bases are the same (both are 3), we can set the exponents equal to each other:

2x = 6x - 8

Now we have a simple linear equation to solve for x. Let's move on to the next step and isolate x.

Step 4: Solve for x

To solve the linear equation 2x = 6x - 8, we first want to get all the x terms on one side. Subtract 6x from both sides:

2x - 6x = -8

-4x = -8

Now, divide both sides by -4 to isolate x:

x* = (-8) / (-4)

x* = 2

So, we have found the value of x that satisfies the equation. To be absolutely sure, it’s always a good idea to check our solution by plugging it back into the original equation.

Verification of the Solution

To ensure that our solution x = 2 is correct, we'll substitute it back into the original equation 3^(2x) = 9^(3x-4) and see if both sides of the equation are equal. This step is crucial because it helps us catch any potential errors we might have made during the solving process.

Step 1: Substitute x = 2 into the original equation

Replace x with 2 in the equation:

3^(2*(2)) = 9^(3*(2)-4)

Step 2: Simplify both sides

Simplify the exponents on both sides:

3^4 = 9^(6-4)

3^4 = 9^2

Now, calculate the values:

81 = 81

Step 3: Check for equality

Since both sides of the equation are equal, our solution x = 2 is correct. This verification step confirms that we have solved the equation accurately and that our value for x satisfies the original condition.

Verifying the solution is an important habit to develop when solving mathematical problems. It not only ensures accuracy but also deepens your understanding of the problem and the solution process. In this case, by verifying that x = 2 indeed makes the original equation true, we have increased confidence in our answer and our ability to solve similar problems.

Conclusion

In this article, we've walked through the process of solving the exponential equation 3^(2x) = 9^(3x-4). We started by understanding the basic principles of exponential equations and the properties that allow us to manipulate and solve them. We then broke down the problem into manageable steps:

1. Rewriting the bases to be the same.
2. Applying the power of a power rule.
3. Equating the exponents.
4. Solving the resulting linear equation.
5. Verifying the solution.

By following these steps, we found that x = 2 is the solution to the equation. The verification step confirmed the accuracy of our solution, reinforcing the importance of checking our work. Mastering these techniques for solving exponential equations is a valuable skill in mathematics and has applications in various real-world scenarios. Understanding exponential growth and decay is fundamental in fields such as finance, biology, and physics.

Remember, the key to solving exponential equations often lies in manipulating the bases and exponents to create a simpler equation. Practice and familiarity with exponent rules are essential for success. By consistently applying these principles, you’ll become more confident and proficient in solving a wide range of exponential problems. Keep exploring, keep practicing, and you'll continue to enhance your mathematical abilities.

For further exploration and practice on exponential equations, you can visit resources like Khan Academy's Exponential Equations Section. This link provides additional examples, exercises, and videos to deepen your understanding.