Solving $(\sqrt{7})^{6x} = 49^{x-6}$ An Exponential Equation

Introduction

In this article, we delve into the process of solving an exponential equation, specifically $(\sqrt{7})^{6x} = 49^{x-6}$. Exponential equations, where the variable appears in the exponent, are a fundamental concept in algebra and calculus. Mastering the techniques to solve these equations is crucial for various mathematical applications. This article aims to provide a comprehensive, step-by-step solution, ensuring clarity and understanding for readers of all backgrounds. We will explore the properties of exponents and logarithms, which are essential tools in manipulating and simplifying exponential expressions. By the end of this discussion, you will have a solid grasp of how to tackle similar problems and a deeper appreciation for the elegance of exponential equations.

Understanding Exponential Equations

Exponential equations involve expressions where a constant base is raised to a variable exponent. The key to solving these equations lies in manipulating them so that both sides have the same base or by using logarithms to bring the exponents down. Our specific equation, $(\sqrt{7})^{6x} = 49^{x-6}$, presents an interesting challenge as the bases, $\sqrt{7}$ and $49$, are related but not immediately the same. This is where our understanding of exponent properties comes into play. We will leverage the fact that $49$ is a power of $7$ and that $\sqrt{7}$ is also a power of $7$. By expressing both sides of the equation with the same base, we can equate the exponents and solve for $x$. This method not only simplifies the problem but also provides a clear pathway to the solution. The beauty of this approach lies in its simplicity and directness, showcasing the power of strategic manipulation in mathematical problem-solving. Furthermore, understanding the relationship between different bases and their exponents is a cornerstone of more advanced mathematical concepts, such as exponential growth and decay, which are prevalent in fields like physics, finance, and biology. Therefore, mastering this skill is not just about solving a single equation but about building a foundation for future mathematical endeavors.

The Role of Logarithms

While we will solve this particular equation by making the bases the same, it's essential to acknowledge the role of logarithms in solving exponential equations. Logarithms provide an alternative approach, especially when it's not straightforward to express both sides with the same base. A logarithm is the inverse operation to exponentiation. If $a^b = c$, then $\log_a c = b$. This inverse relationship allows us to "bring down" the exponent, transforming an exponential equation into a more manageable algebraic equation. For instance, if we had an equation like $2^x = 5$, we could take the logarithm of both sides (using any base, but commonly base 10 or base $e$, the natural logarithm) to solve for $x$. The logarithmic approach is particularly useful when dealing with irrational exponents or bases that don't have a simple relationship. However, for the equation at hand, $(\sqrt{7})^{6x} = 49^{x-6}$, the method of equating bases is more direct and efficient. We will use the properties of exponents to rewrite the equation in a form where both sides have the same base, allowing us to directly compare the exponents. This showcases the versatility in solving exponential equations and the importance of choosing the most appropriate method based on the specific problem. Understanding both the method of equating bases and the logarithmic approach provides a comprehensive toolkit for tackling a wide range of exponential equations.

Step-by-Step Solution

Step 1: Express Both Sides with the Same Base

Our primary goal is to express both sides of the equation $(\sqrt{7})^{6x} = 49^{x-6}$ with the same base. This will allow us to equate the exponents and solve for $x$. We recognize that both $\sqrt{7}$ and $49$ can be expressed as powers of $7$. Specifically, $\sqrt{7}$ is $7^{\frac{1}{2}}$, and $49$ is $7^2$. This is a crucial observation, as it provides the foundation for simplifying the equation. Substituting these expressions into the original equation, we get $(7^{\frac{1}{2}})^{6x} = (7^2)^{x-6}$. This transformation is a key step in solving exponential equations. By identifying a common base, we can leverage the properties of exponents to simplify the equation significantly. The next step involves applying the power of a power rule, which states that $(a^m)^n = a^{mn}$. This rule will allow us to further simplify both sides of the equation and move closer to isolating $x$. The strategic use of exponent properties is a fundamental skill in algebra, enabling us to manipulate complex expressions into simpler, more manageable forms. This step demonstrates the importance of recognizing relationships between numbers and their powers, a skill that is invaluable in various mathematical contexts.

Step 2: Apply the Power of a Power Rule

Now, we apply the power of a power rule, $(a^m)^n = a^{mn}$, to both sides of the equation $(7^{\frac{1}{2}})^{6x} = (7^2)^{x-6}$. On the left side, we have $(7^{\frac{1}{2}})^{6x}$, which simplifies to $7^{(\frac{1}{2} imes 6x)} = 7^{3x}$. On the right side, we have $(7^2)^{x-6}$, which simplifies to $7^{2(x-6)} = 7^{2x-12}$. The equation now becomes $7^{3x} = 7^{2x-12}$. This simplified form is a direct consequence of the power of a power rule, which is a cornerstone of exponent manipulation. By applying this rule, we have effectively eliminated the parentheses and expressed both sides of the equation as powers of $7$. This simplification is a critical step towards solving for $x$, as it allows us to focus solely on the exponents. The next logical step is to equate the exponents, since the bases are now the same. This is a powerful technique in solving exponential equations, as it transforms the problem from an exponential one to a simple algebraic one. The ability to recognize and apply exponent rules is a fundamental skill in algebra, and this step highlights its importance in simplifying and solving equations.

Step 3: Equate the Exponents

With the equation now in the form $7^{3x} = 7^{2x-12}$, we can equate the exponents. This is a valid step because if $a^m = a^n$, then $m = n$. Therefore, we have $3x = 2x - 12$. This transformation is the heart of solving exponential equations when the bases are the same. By equating the exponents, we've converted the exponential equation into a linear equation, which is much easier to solve. This step highlights the elegance of mathematical problem-solving, where a seemingly complex problem is reduced to a simpler one through strategic manipulation. The equation $3x = 2x - 12$ is a straightforward linear equation that can be solved using basic algebraic techniques. The next step involves isolating $x$ by performing algebraic operations on both sides of the equation. This process demonstrates the interconnectedness of different mathematical concepts, as we transition from exponential equations to linear equations within a single problem.

Step 4: Solve for $x$

Now, we solve the linear equation $3x = 2x - 12$ for $x$. To isolate $x$, we subtract $2x$ from both sides of the equation, which gives us $3x - 2x = 2x - 12 - 2x$, simplifying to $x = -12$. This is the solution to the equation. We have successfully found the value of $x$ that satisfies the original exponential equation. This final step demonstrates the culmination of our efforts, where the strategic application of exponent properties and algebraic manipulation has led us to the solution. It's always a good practice to check the solution by substituting it back into the original equation to ensure its validity. In this case, substituting $x = -12$ into $(\sqrt{7})^{6x} = 49^{x-6}$ will confirm that both sides of the equation are equal. This verification step provides confidence in the accuracy of the solution and reinforces the understanding of the problem-solving process. The solution $x = -12$ is a negative value, which is perfectly acceptable in the context of exponential equations. This problem showcases the power of algebraic techniques in solving complex equations and highlights the importance of a systematic approach to problem-solving.

Conclusion

In conclusion, we have successfully solved the exponential equation $(\sqrt{7})^{6x} = 49^{x-6}$ by expressing both sides with the same base, applying the power of a power rule, equating the exponents, and solving the resulting linear equation. The solution we found is $x = -12$. This problem demonstrates the importance of understanding and applying the properties of exponents, as well as the ability to manipulate algebraic expressions. The strategic use of these skills allowed us to transform a seemingly complex equation into a simple one, highlighting the elegance and power of mathematical problem-solving. The techniques used in this solution can be applied to a wide range of exponential equations, making it a valuable tool in your mathematical arsenal. Mastering these techniques not only enhances your ability to solve equations but also deepens your understanding of the fundamental principles of algebra. The process of solving exponential equations reinforces the importance of a systematic approach, where each step is carefully considered and executed. This approach is not only beneficial in mathematics but also in various other fields where problem-solving is essential. By understanding the underlying principles and practicing regularly, you can confidently tackle more challenging problems and expand your mathematical horizons.