Mastering Exponential Equations A Step-by-Step Guide

To select the correct answer when faced with a logarithmic equation and asked to find its equivalent exponential form, a clear understanding of the relationship between logarithms and exponentials is crucial. This article will delve into the intricacies of this transformation, providing a step-by-step approach to solving the given problem and offering a comprehensive guide to mastering these mathematical concepts. Let's explore how to convert the logarithmic equation $\log_7 6 + 3\log_7 2 = x$ into its equivalent exponential form. Understanding the fundamentals of logarithms and exponential functions is essential not only for solving this specific problem but also for tackling a wide range of mathematical challenges. The ability to fluently move between logarithmic and exponential forms is a cornerstone of advanced mathematical thinking, with applications spanning fields such as calculus, physics, and computer science. In this detailed exploration, we will break down each step involved in the transformation process, ensuring that you gain a solid grasp of the underlying principles and can confidently apply them in various contexts. By the end of this guide, you will have not only the solution to the given equation but also a broader understanding of how logarithms and exponentials interact, empowering you to solve more complex problems in the future.

Understanding Logarithmic Equations

Logarithmic equations are a powerful way to express relationships where a base raised to a certain power equals a given number. The equation $\log_b a = c$ is read as "the logarithm of a to the base b is c" and is equivalent to the exponential form $b^c = a$. Here, b is the base, a is the argument, and c is the exponent or logarithm. Before we dive into solving the specific problem at hand, it's crucial to have a firm grasp of the basic properties of logarithms. These properties are the tools that will allow us to manipulate logarithmic expressions and transform them into more manageable forms. One of the most important properties is the power rule, which states that $\log_b (x^p) = p \log_b x$. This rule allows us to move exponents within a logarithm, which is a critical step in simplifying complex logarithmic expressions. Another key property is the product rule, which states that $\log_b (xy) = \log_b x + \log_b y$. This rule enables us to combine logarithms of products into sums of logarithms, and vice versa. Conversely, the quotient rule states that $\log_b (x/y) = \log_b x - \log_b y$, allowing us to handle division within logarithms. Understanding these properties is not just about memorizing formulas; it's about grasping the underlying principles that govern how logarithms behave. When you truly understand these principles, you can apply them flexibly and creatively to solve a wide range of problems. This foundational knowledge is what will allow you to tackle more complex logarithmic equations and to confidently navigate the world of mathematics.

Applying Logarithmic Properties

To solve the equation $\log_7 6 + 3\log_7 2 = x$, we first need to simplify the left-hand side using logarithmic properties. The key property to apply here is the power rule, which allows us to rewrite the term $3\log_7 2$. According to the power rule, $3\log_7 2$ is equivalent to $\log_7 (2^3)$. So, the equation becomes $\log_7 6 + \log_7 (2^3) = x$. Next, we simplify $2^3$ to get 8, so our equation now looks like this: $\log_7 6 + \log_7 8 = x$. To further simplify, we can use the product rule of logarithms. The product rule states that the sum of two logarithms with the same base is equal to the logarithm of the product of their arguments. In other words, $\log_b m + \log_b n = \log_b (mn)$. Applying this rule to our equation, we get $\log_7 (6 \times 8) = x$. Multiplying 6 and 8 gives us 48, so the equation simplifies to $\log_7 48 = x$. Now we have a single logarithm equal to x, which is a much more manageable form. This step-by-step simplification process is crucial for solving logarithmic equations. By breaking down the problem into smaller, more digestible steps, we can avoid making errors and gain a clearer understanding of the underlying mathematical principles. The ability to manipulate logarithmic expressions using these properties is a fundamental skill in mathematics, and mastering it will open doors to solving a wide range of problems.

Converting to Exponential Form

Now that we have simplified the logarithmic equation to $\log_7 48 = x$, the next step is to convert it to its equivalent exponential form. The fundamental relationship between logarithms and exponentials is that if $\log_b a = c$, then $b^c = a$. This relationship is the key to transforming between logarithmic and exponential expressions. In our equation, $\log_7 48 = x$, we can identify the base as 7, the argument as 48, and the logarithm as x. Applying the conversion formula, we raise the base (7) to the power of the logarithm (x) and set it equal to the argument (48). This gives us the exponential equation $7^x = 48$. This transformation is a direct application of the definition of a logarithm and highlights the inverse relationship between logarithmic and exponential functions. Understanding this relationship is crucial for solving a wide range of mathematical problems, from simple equations to complex applications in calculus and other advanced fields. By converting the logarithmic equation into its exponential form, we have effectively re-expressed the same relationship in a different way. This new form can be more convenient for certain types of calculations or for gaining a different perspective on the problem. The ability to seamlessly switch between logarithmic and exponential forms is a powerful tool in the mathematician's toolkit, allowing for greater flexibility and problem-solving efficiency.

Identifying the Correct Answer

After converting the logarithmic equation $\log_7 6 + 3\log_7 2 = x$ to its exponential form, we found that $7^x = 48$. Now, let's review the options provided: A. $7^x = 48$ B. $x^7 = 12$ C. $7^x = 12$ D. $x^7 = 48$ Comparing our result, $7^x = 48$, with the given options, we can clearly see that option A, $7^x = 48$, matches our solution. Therefore, option A is the correct answer. Options B, C, and D do not match the exponential form we derived and can be eliminated. This process of comparing the derived solution with the given options is a critical step in any problem-solving scenario. It ensures that we have not made any errors in our calculations and that we have correctly interpreted the question. In this case, the correct answer directly matches the exponential form we derived, reinforcing our confidence in the solution. This step also highlights the importance of careful attention to detail when working with mathematical equations. A small error in the conversion or simplification process could lead to an incorrect answer, so it's always wise to double-check your work and compare your result with the given options.

Conclusion

In summary, the exponential equation equivalent to $\log_7 6 + 3\log_7 2 = x$ is $7^x = 48$. This was achieved by first applying the power rule of logarithms to simplify $3\log_7 2$ to $\log_7 2^3$, which equals $\log_7 8$. Then, we used the product rule to combine $\log_7 6 + \log_7 8$ into $\log_7 (6 \times 8)$, resulting in $\log_7 48 = x$. Finally, we converted the logarithmic equation to its exponential form, $7^x = 48$. This comprehensive step-by-step solution demonstrates the importance of understanding and applying logarithmic properties, as well as the fundamental relationship between logarithmic and exponential forms. Mastering these concepts is crucial for success in mathematics, as they are foundational to more advanced topics such as calculus and differential equations. By working through this problem and understanding the underlying principles, you have not only solved a specific equation but also gained valuable skills that will serve you well in your future mathematical endeavors. Remember, mathematics is a cumulative subject, and each concept builds upon the previous one. By taking the time to thoroughly understand the basics, you will be well-equipped to tackle more complex challenges and achieve your mathematical goals.