Solving For X In 3^(2x) = 9^(3x-4) A Step-by-Step Guide

Jul 11, 2025 by Admin 56 views

Solving Exponential Equations: Finding the Value of x in 3^(2x) = 9^(3x-4)

In this article, we will delve into the process of solving exponential equations, focusing on the equation 3^(2x) = 9^(3x-4). This type of problem is a staple in algebra and requires a solid understanding of exponential properties and equation manipulation. We will break down the steps involved in finding the value of x that satisfies this equation, providing a clear and comprehensive guide for students and math enthusiasts alike. Understanding how to solve exponential equations is crucial for various applications in mathematics, science, and engineering, making this a valuable skill to master.

Understanding Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. To effectively solve these equations, it's essential to grasp the fundamental properties of exponents. One of the key strategies in solving exponential equations is to express both sides of the equation with the same base. This allows us to equate the exponents and solve for the variable. In the given equation, 3^(2x) = 9^(3x-4), we can observe that 9 is a power of 3 (9 = 3^2). This observation is crucial for simplifying the equation and bringing it to a solvable form. Recognizing these relationships and applying the properties of exponents are the cornerstones of solving exponential equations efficiently.

Rewriting the Equation with a Common Base

The first step in solving the equation 3^(2x) = 9^(3x-4) is to rewrite both sides with a common base. Since 9 can be expressed as 3^2, we can rewrite the equation as:

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

This step is crucial because it allows us to use the property of exponents that states (a^m)n = a^(mn)*. Applying this property to the right side of the equation, we get:

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

Now, both sides of the equation have the same base (3), which sets the stage for equating the exponents. This transformation is a fundamental technique in solving exponential equations and simplifies the problem significantly. By expressing both sides with the same base, we can eliminate the exponential part and focus on the algebraic manipulation of the exponents.

Equating the Exponents

With both sides of the equation expressed with the same base, we can now equate the exponents. From the equation 3^(2x) = 3^(2(3x-4))*, we can deduce that the exponents must be equal:

2x = 2(3x - 4)

This step is a direct consequence of the fact that exponential functions are one-to-one. This means that if a^m = a^n, then m = n, provided that a is a positive number not equal to 1. By equating the exponents, we transform the exponential equation into a simple linear equation, which is much easier to solve. This is a key step in solving exponential equations, as it allows us to apply standard algebraic techniques to find the value of the variable.

Solving the Linear Equation

After equating the exponents, we are left with a linear equation: 2x = 2(3x - 4). The next step is to solve this equation for x. First, we distribute the 2 on the right side of the equation:

2x = 6x - 8

Next, we want to isolate the x terms on one side of the equation. We can subtract 6x from both sides:

2x - 6x = -8

This simplifies to:

-4x = -8

Finally, we divide both sides by -4 to solve for x:

x = (-8) / (-4)
x = 2

Therefore, the value of x that satisfies the equation 2x = 2(3x - 4) is 2. This step-by-step solution demonstrates the process of solving a linear equation, a fundamental skill in algebra. By carefully applying algebraic operations, we can isolate the variable and find its value.

Verifying the Solution

After finding a solution to an equation, it's always a good practice to verify the solution by substituting it back into the original equation. This ensures that the solution is correct and that no errors were made during the solving process. In our case, the original equation is 3^(2x) = 9^(3x-4), and we found that x = 2. Let's substitute x = 2 into the original equation:

3^(2*2) = 9^(3*2-4)
3^4 = 9^(6-4)
3^4 = 9^2

Now, we evaluate both sides:

81 = 81

Since both sides of the equation are equal, our solution x = 2 is correct. This verification step is crucial in ensuring the accuracy of the solution and provides confidence in the result. By substituting the solution back into the original equation, we can confirm that it satisfies the equation and that no algebraic errors were made.

Final Answer

Therefore, the value of x that satisfies the equation 3^(2x) = 9^(3x-4) is 2. We arrived at this solution by rewriting the equation with a common base, equating the exponents, solving the resulting linear equation, and verifying the solution. This step-by-step process demonstrates a systematic approach to solving exponential equations. Understanding these techniques is crucial for mastering algebra and its applications in various fields.

Alternative Methods for Solving Exponential Equations

While the method we used is the most straightforward for this particular problem, there are other techniques that can be used to solve exponential equations. One common method involves the use of logarithms. Logarithms are particularly useful when it is not possible to express both sides of the equation with the same base. For instance, if we had an equation like 3^x = 5, we could take the logarithm of both sides to solve for x. However, in our case, since we could express both sides with the same base, using logarithms was not necessary.

Another approach involves graphical methods. We could graph the functions y = 3^(2x) and y = 9^(3x-4) and find the point of intersection. The x-coordinate of the intersection point would be the solution to the equation. Graphical methods are particularly useful for visualizing the solutions and for equations that are difficult to solve algebraically.

Common Mistakes to Avoid

When solving exponential equations, there are several common mistakes that students often make. One common mistake is failing to rewrite the equation with a common base before equating the exponents. This can lead to incorrect solutions. Another mistake is incorrectly applying the properties of exponents. For instance, students might incorrectly simplify (3²⁾(3x-4) as 3^(2+3x-4) instead of 3^(2(3x-4))*. It's crucial to carefully apply the properties of exponents to avoid such errors.

Another common mistake is failing to verify the solution. Substituting the solution back into the original equation is a crucial step in ensuring the accuracy of the solution. By avoiding these common mistakes and carefully applying the correct techniques, students can successfully solve exponential equations.

Real-World Applications of Exponential Equations

Exponential equations are not just abstract mathematical concepts; they have numerous real-world applications. They are used to model phenomena such as population growth, radioactive decay, compound interest, and the spread of diseases. For example, in finance, exponential equations are used to calculate the future value of an investment with compound interest. In biology, they are used to model the growth of bacterial populations. In physics, they are used to describe the decay of radioactive substances. Understanding exponential equations is therefore essential for anyone working in these fields.

The equation 3^(2x) = 9^(3x-4), while a specific example, illustrates the general principles involved in solving exponential equations. By mastering these principles, students can tackle a wide range of problems in mathematics and its applications. The ability to solve exponential equations is a valuable skill that opens doors to further study in mathematics, science, and engineering. Through practice and a solid understanding of the underlying concepts, anyone can become proficient in solving exponential equations and applying them to real-world problems.

In conclusion, solving the equation 3^(2x) = 9^(3x-4) involves a series of steps that require a solid understanding of exponential properties and algebraic manipulation. By rewriting the equation with a common base, equating the exponents, solving the resulting linear equation, and verifying the solution, we can find the value of x that satisfies the equation. This process not only provides a solution to the specific problem but also illustrates the general techniques used to solve exponential equations, which have wide-ranging applications in various fields.