Solving Logarithmic Equations A Step-by-Step Guide To 3 Log₄ X = Log₄ 32 + Log₄ 2

Introduction to Logarithmic Equations

In the realm of mathematics, logarithmic equations hold a significant position, particularly in fields like calculus, physics, and engineering. These equations involve logarithms, which are the inverse functions of exponential functions. Understanding logarithmic equations is crucial for solving problems related to exponential growth, decay, and various scientific calculations. This article delves into the step-by-step solution of the logarithmic equation: 3 log₄ x = log₄ 32 + log₄ 2, providing a comprehensive guide for students and enthusiasts alike. We will explore the fundamental properties of logarithms, apply them to simplify the equation, and ultimately find the value of x that satisfies the given condition. By the end of this guide, you will have a clear understanding of how to tackle similar logarithmic equations with confidence and precision.

Before diving into the solution, let's establish a solid foundation by understanding the basic properties of logarithms. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Mathematically, if b^y = x, then log_b(x) = y, where b is the base, x is the argument, and y is the logarithm. Common bases include 10 (common logarithm) and e (natural logarithm), but in our equation, we're dealing with base 4 logarithms. Several key properties of logarithms will be instrumental in solving our equation, including the product rule, the quotient rule, and the power rule. The product rule states that the logarithm of the product of two numbers is the sum of the logarithms of the individual numbers (log_b(mn) = log_b(m) + log_b(n)). The quotient rule states that the logarithm of the quotient of two numbers is the difference of their logarithms (log_b(m/n) = log_b(m) - log_b(n)). The power rule states that the logarithm of a number raised to a power is the product of the power and the logarithm of the number (log_b(m^p) = p log_b(m)). These properties allow us to manipulate logarithmic expressions and simplify equations, making them easier to solve. In the following sections, we will apply these properties to the equation at hand, systematically reducing it to a form where we can isolate x and find its value. The ability to recognize and apply these logarithmic properties is a cornerstone of mathematical proficiency, opening doors to solving complex problems across various scientific and engineering disciplines. So, let's embark on this journey to unravel the solution to our logarithmic equation, armed with the knowledge of these fundamental principles.

Step 1: Simplify the Right Side of the Equation

Our first step in solving the equation 3 log₄ x = log₄ 32 + log₄ 2 is to simplify the right side. We can accomplish this by utilizing the product rule of logarithms. The product rule, as mentioned earlier, states that log_b(mn) = log_b(m) + log_b(n). In our case, we have the sum of two logarithms with the same base (base 4), so we can combine them into a single logarithm by multiplying their arguments. Applying the product rule to log₄ 32 + log₄ 2, we get log₄ (32 * 2). Multiplying 32 and 2 gives us 64, so the expression simplifies to log₄ 64. Now, we need to determine what power we must raise 4 to in order to get 64. In other words, we need to find y such that 4^y = 64. We can express 64 as a power of 4: 64 = 4^3. Therefore, log₄ 64 = 3. By simplifying the right side of the equation to 3, we have made significant progress in solving for x. The equation now reads 3 log₄ x = 3, which is a much simpler form to work with. This step highlights the power of logarithmic properties in transforming complex expressions into more manageable ones. By recognizing and applying the product rule, we have reduced the equation to a form where we can focus on isolating x and finding its value. The ability to manipulate logarithmic expressions in this way is a crucial skill for anyone working with logarithmic equations. In the next step, we will further simplify the equation by dealing with the coefficient on the left side, paving the way for the final solution.

This initial simplification is crucial not only for solving this specific equation but also for developing a general strategy for approaching logarithmic problems. Recognizing patterns and applying the appropriate logarithmic properties is a skill that grows with practice. The product rule, in particular, is one of the most frequently used properties, and mastering its application is essential. By carefully breaking down the problem into smaller, manageable steps, we can tackle even the most complex logarithmic equations. This methodical approach not only leads to the correct solution but also enhances our understanding of the underlying principles of logarithms. As we move forward, we will continue to employ this step-by-step strategy, ensuring that each step is clearly explained and justified. This will not only help in solving the current equation but also provide a solid foundation for tackling future logarithmic challenges. The simplification achieved in this first step demonstrates the elegance and power of logarithmic properties in making complex problems accessible and solvable. With the right side of the equation now simplified to 3, we are well-positioned to proceed with the next steps and ultimately determine the value of x.

Step 2: Isolate the Logarithm

Having simplified the right side of the equation, our next objective is to isolate the logarithm term on the left side. Currently, the equation stands as 3 log₄ x = 3. The logarithm term, log₄ x, is being multiplied by the coefficient 3. To isolate log₄ x, we need to undo this multiplication. The inverse operation of multiplication is division, so we will divide both sides of the equation by 3. Dividing both sides by 3, we get (3 log₄ x) / 3 = 3 / 3. This simplifies to log₄ x = 1. Now, we have successfully isolated the logarithm term. This step is a crucial milestone in solving the equation, as it brings us closer to finding the value of x. Isolating the logarithmic term allows us to directly relate x to the base and the logarithm's value, setting the stage for converting the equation into exponential form. The principle of performing the same operation on both sides of an equation is fundamental to maintaining equality and solving for unknowns. In this case, dividing both sides by 3 ensures that the equation remains balanced while simultaneously simplifying it. This principle applies not only to logarithmic equations but also to various other types of equations in algebra and calculus. By carefully applying this principle, we can systematically manipulate equations to isolate the variable of interest.

The equation log₄ x = 1 is now in a form that is much easier to interpret. It states that the logarithm of x to the base 4 is equal to 1. This means that 4 raised to the power of 1 is equal to x. This understanding is key to the next step, where we will convert the logarithmic equation into its equivalent exponential form. The ability to move between logarithmic and exponential forms is a fundamental skill in working with logarithms. It allows us to translate between different perspectives on the same relationship, making it easier to solve for unknowns. The isolation of the logarithm term is not just a mechanical step; it represents a conceptual shift in how we view the equation. By isolating log₄ x, we are focusing on the relationship between x and the base 4, which is essential for finding the value of x. This step-by-step approach to problem-solving is a hallmark of mathematical thinking. By breaking down a complex problem into smaller, manageable steps, we can systematically arrive at the solution. The isolation of the logarithm term is a prime example of this approach, demonstrating how a seemingly complex equation can be simplified through careful manipulation. With the logarithm term now isolated, we are ready to take the final step and solve for x.

Step 3: Convert to Exponential Form and Solve for x

With the equation now in the simplified form log₄ x = 1, we are ready to convert it into its equivalent exponential form. This conversion is the key to unlocking the value of x. Recall the fundamental relationship between logarithms and exponentials: if log_b(x) = y, then b^y = x. Applying this relationship to our equation, where the base b is 4, the logarithm x is the unknown, and the result y is 1, we can rewrite the equation as 4^1 = x. This conversion is a direct application of the definition of logarithms. It translates the logarithmic statement "the power to which we must raise 4 to get x is 1" into the exponential statement "4 raised to the power of 1 equals x." Now, solving for x is straightforward. We know that any number raised to the power of 1 is simply the number itself. Therefore, 4^1 = 4. This means that x = 4. We have successfully found the value of x that satisfies the original equation. This step highlights the power of understanding the fundamental relationship between logarithms and exponentials. By converting the logarithmic equation into its equivalent exponential form, we were able to directly solve for x. This conversion is a crucial skill for anyone working with logarithmic equations, as it provides a direct pathway to the solution.

The conversion to exponential form not only allows us to solve for x but also provides a deeper understanding of the equation. It reinforces the idea that logarithms and exponentials are inverse operations, each undoing the effect of the other. This understanding is essential for tackling more complex logarithmic problems. The solution x = 4 means that if we substitute 4 back into the original equation, it will hold true. This is a crucial step in verifying our solution. To verify, we substitute x = 4 into the original equation: 3 log₄ (4) = log₄ 32 + log₄ 2. Since log₄ (4) = 1, the left side of the equation becomes 3 * 1 = 3. As we found earlier, log₄ 32 + log₄ 2 simplifies to log₄ 64, which is equal to 3. Therefore, the right side of the equation is also 3. Since both sides of the equation are equal when x = 4, we have verified that our solution is correct. This verification step is an important part of the problem-solving process. It helps to catch any errors that may have occurred during the solution process and ensures that the final answer is accurate. In conclusion, by converting the logarithmic equation to exponential form and solving for x, we have successfully found the solution to the equation 3 log₄ x = log₄ 32 + log₄ 2. The solution is x = 4. This process demonstrates the power of logarithmic properties and the importance of understanding the relationship between logarithms and exponentials. With a solid understanding of these concepts, we can confidently tackle a wide range of logarithmic equations.

Conclusion

In summary, we have successfully solved the logarithmic equation 3 log₄ x = log₄ 32 + log₄ 2 by systematically applying logarithmic properties and converting the equation into exponential form. The solution process involved three key steps: simplifying the right side of the equation using the product rule of logarithms, isolating the logarithm term on the left side by dividing both sides by 3, and converting the equation to exponential form to solve for x. Through these steps, we determined that x = 4 is the solution to the equation. This exercise has demonstrated the importance of understanding and applying logarithmic properties, such as the product rule, and the fundamental relationship between logarithms and exponentials. The ability to manipulate logarithmic expressions and equations is a crucial skill in mathematics, with applications in various fields, including science, engineering, and finance. By mastering these techniques, students and professionals can confidently tackle a wide range of problems involving logarithmic functions.

The step-by-step approach we employed in solving this equation is a valuable problem-solving strategy that can be applied to other mathematical challenges as well. Breaking down a complex problem into smaller, more manageable steps allows us to systematically address each aspect of the problem and arrive at the solution. This approach not only helps in finding the correct answer but also enhances our understanding of the underlying concepts. The verification step, where we substituted the solution back into the original equation, is also an essential part of the problem-solving process. It ensures that our solution is accurate and helps to catch any errors that may have occurred during the solution process. Logarithmic equations are a fundamental topic in mathematics, and mastering their solution is essential for further studies in calculus and other advanced mathematical topics. The concepts and techniques discussed in this article provide a solid foundation for tackling more complex logarithmic problems. By understanding the properties of logarithms and the relationship between logarithms and exponentials, we can confidently approach a wide range of mathematical challenges. The solution to the equation 3 log₄ x = log₄ 32 + log₄ 2 serves as a testament to the power of mathematical reasoning and the importance of a systematic approach to problem-solving. With practice and a solid understanding of the fundamental principles, we can successfully navigate the world of logarithmic equations and their applications.