Solving Exponential Equations A Step-by-Step Guide For 2^(4x-2) = 6^x

In this article, we will delve into solving the exponential equation 2^(4x-2) = 6^x. Exponential equations, where the variable appears in the exponent, are a fundamental concept in mathematics. These equations are encountered in various fields, including calculus, physics, engineering, and economics. The solution to this particular equation involves the application of logarithmic properties to isolate the variable x. This comprehensive guide aims to provide a step-by-step approach to solving the given exponential equation, ensuring clarity and understanding for readers of all mathematical backgrounds. We will begin by explaining the basic principles of exponential equations and logarithms, then move on to the detailed steps for solving the equation. Furthermore, we will discuss the importance of checking the solution and provide additional insights into similar problem-solving strategies. Understanding how to solve exponential equations is crucial for anyone studying advanced mathematics or working in fields that rely on mathematical modeling. The methods discussed here can be applied to a wide range of exponential equations, making this a valuable skill for students and professionals alike. By the end of this article, you will have a clear understanding of the techniques involved and be confident in your ability to solve similar problems. So, let's embark on this mathematical journey and unravel the solution to 2^(4x-2) = 6^x.

Understanding Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. These equations often arise in modeling real-world phenomena such as population growth, radioactive decay, and compound interest. Solving them requires a solid understanding of exponential and logarithmic functions, as well as their properties. Exponential functions are characterized by rapid growth or decay, making them powerful tools for representing various natural processes. The general form of an exponential equation is a^(f(x)) = b^(g(x)), where a and b are constants, and f(x) and g(x) are functions of x. To solve such equations, we often need to use logarithms, which are the inverse functions of exponentials. Logarithms allow us to bring the exponent down as a coefficient, thereby transforming the exponential equation into a more manageable algebraic form. The key properties of logarithms, such as the power rule, product rule, and quotient rule, are essential in this process. Moreover, understanding the relationship between exponential and logarithmic functions is crucial for solving not only simple exponential equations but also more complex problems involving exponential models. In this context, we will use these principles to address the specific equation 2^(4x-2) = 6^x. The detailed steps below will illustrate how these properties are applied to isolate x and find the solution. By mastering these techniques, you'll gain the ability to tackle a wide variety of exponential equations and related mathematical challenges.

Steps to Solve 2^(4x-2) = 6^x

To solve the exponential equation 2^(4x-2) = 6^x, we will employ logarithmic properties. Logarithms are the inverse operation to exponentiation, and they are instrumental in solving equations where the variable is in the exponent. Here is a step-by-step breakdown of the solution:

Step 1: Apply Logarithms to Both Sides

The first step is to apply a logarithm to both sides of the equation. This allows us to use the logarithmic property that helps bring the exponent down as a coefficient. We can use any base for the logarithm, but the common logarithm (base 10) or the natural logarithm (base e) are often the most convenient. In this case, we will use the natural logarithm (ln), but the process would be similar with the common logarithm. Applying the natural logarithm to both sides, we get:

ln(2^(4x-2)) = ln(6^x)

This step is crucial as it sets the stage for using logarithmic properties to simplify the equation. By taking the logarithm of both sides, we convert the exponential equation into a form where we can apply the power rule of logarithms, which will be used in the next step.

Step 2: Use the Power Rule of Logarithms

The power rule of logarithms states that ln(a^b) = b * ln(a). Applying this rule to both sides of the equation, we get:

(4x - 2) * ln(2) = x * ln(6)

This transformation is a key step in solving the exponential equation. By using the power rule, we have successfully brought the exponents down as coefficients, turning the exponential equation into a linear equation in terms of x. This makes the equation much easier to manipulate and solve. The next step involves distributing the terms and rearranging the equation to isolate x.

Step 3: Distribute and Rearrange the Equation

Next, we distribute ln(2) on the left side of the equation:

4x * ln(2) - 2 * ln(2) = x * ln(6)

Now, we want to get all terms involving x on one side and constants on the other. Subtract x * ln(6) from both sides:

4x * ln(2) - x * ln(6) = 2 * ln(2)

This step is essential for isolating x. By rearranging the terms, we are setting up the equation to be factored and solved for x. The next step will involve factoring out x from the terms on the left side of the equation.

Step 4: Factor out x

Factor out x from the left side:

x * (4 * ln(2) - ln(6)) = 2 * ln(2)

This factorization is a crucial step in isolating x. By factoring x out, we can now easily solve for x by dividing both sides of the equation by the expression in parentheses. The next step involves this division to find the value of x.

Step 5: Solve for x

Divide both sides by (4 * ln(2) - ln(6)) to isolate x:

x = (2 * ln(2)) / (4 * ln(2) - ln(6))

This is the exact solution for x. However, to get a decimal approximation, we need to use a calculator to evaluate the expression. The next step involves using a calculator to find the numerical value of x.

Step 6: Approximate the Value of x

Using a calculator, we approximate the value of x:

x ≈ (2 * 0.6931) / (4 * 0.6931 - 1.7918)

x ≈ 1.3862 / (2.7724 - 1.7918)

x ≈ 1.3862 / 0.9806

x ≈ 1.413

Rounding to three decimal places, we get x ≈ 1.413.

Final Answer

The solution to the exponential equation 2^(4x-2) = 6^x is approximately x ≈ 1.413. This result is obtained by applying logarithms, using the power rule, and performing algebraic manipulations to isolate x. The final approximation is found by using a calculator to evaluate the logarithmic expressions.

Verifying the Solution

Verifying the solution is a crucial step in solving any equation, especially exponential equations, to ensure the accuracy of the result. To verify our solution x ≈ 1.413 for the equation 2^(4x-2) = 6^x, we will substitute this value back into the original equation and check if both sides are approximately equal. This process helps to identify any potential errors made during the solving process, such as incorrect application of logarithmic properties or miscalculations. It also confirms that the approximated value is a valid solution within the context of the equation. By substituting the value, we are essentially testing the equality of the equation with the found solution. If both sides of the equation are nearly equal after the substitution, it provides strong evidence that our solution is correct. This step is not just a formality; it is an integral part of the problem-solving process that enhances confidence in the answer and highlights the importance of precision in mathematical calculations.

Substituting x ≈ 1.413 into the Original Equation

Let's substitute x ≈ 1.413 into the original equation 2^(4x-2) = 6^x:

Left side: 2^(4 * 1.413 - 2) = 2^(5.652 - 2) = 2^3.652

Right side: 6^1.413

Evaluating Both Sides

Now, we use a calculator to evaluate both sides:

Left side: 2^3.652 ≈ 12.60

Right side: 6^1.413 ≈ 12.60

Since both sides are approximately equal (12.60 ≈ 12.60), our solution x ≈ 1.413 is verified.

Common Mistakes to Avoid

When solving exponential equations, there are several common mistakes that students often make. Being aware of these pitfalls can help ensure accurate solutions. One frequent error is incorrectly applying the logarithmic properties. For example, students might mistakenly assume that ln(a + b) is equal to ln(a) + ln(b), which is not true. The correct logarithmic properties must be used precisely to manipulate equations effectively. Another common mistake is failing to distribute terms properly. When an expression like (4x - 2) * ln(2) appears, it is essential to distribute ln(2) to both 4x and -2. Forgetting to do so can lead to an incorrect equation and, consequently, a wrong solution. Additionally, errors can occur when isolating the variable. It's crucial to perform algebraic manipulations carefully, ensuring that terms are moved correctly across the equation. This includes paying attention to signs and performing operations in the correct order. Another significant mistake is neglecting to verify the solution. As demonstrated earlier, verifying the solution by substituting it back into the original equation is a vital step to confirm its validity. This can help catch any errors made during the solving process. Lastly, rounding errors can also lead to inaccuracies. While approximating the final answer, it's best to keep as many decimal places as possible during intermediate calculations and round only at the final step. By being mindful of these common mistakes, students can improve their accuracy and confidence in solving exponential equations.

Conclusion

In conclusion, solving the exponential equation 2^(4x-2) = 6^x involves a systematic approach using logarithmic properties. By applying logarithms to both sides, utilizing the power rule, and carefully manipulating the equation, we successfully found the solution x ≈ 1.413. Verifying the solution is a critical step to ensure accuracy, and understanding common mistakes can help prevent errors in future problem-solving. Mastering these techniques provides a strong foundation for tackling more complex exponential equations and related mathematical concepts. Exponential equations are fundamental in various fields, making the ability to solve them an invaluable skill. Through this comprehensive guide, we have demonstrated the step-by-step process, emphasizing the importance of each step and the underlying mathematical principles. By practicing these methods, you can confidently approach and solve a wide range of exponential equations. Remember, the key to success in mathematics lies in understanding the fundamental concepts and applying them consistently and accurately. This article has equipped you with the necessary tools and knowledge to confidently solve exponential equations, setting you on the path to further mathematical achievements.