Solving Logarithmic Equations A Step-by-Step Guide
Introduction
This article provides a comprehensive guide on solving logarithmic equations, specifically focusing on the equation log r + log 4 = 2. We will break down the steps involved in simplifying the equation and isolating the variable 'r' to find its value. Logarithmic equations are a fundamental concept in mathematics, appearing in various fields such as physics, engineering, and computer science. Understanding how to solve them is crucial for anyone pursuing studies or careers in these areas. This guide aims to provide a clear and concise explanation of the process, making it accessible to students and professionals alike. We will cover the key properties of logarithms, including the product rule, which is essential for simplifying the given equation. Additionally, we will demonstrate how to convert logarithmic equations into exponential form, a technique that allows us to solve for the unknown variable. By the end of this article, you will have a solid understanding of how to solve logarithmic equations and be able to apply these skills to similar problems. The examples provided will illustrate the concepts discussed, ensuring that you grasp the underlying principles. Whether you are a student learning about logarithms for the first time or a professional looking to refresh your knowledge, this guide will serve as a valuable resource. Let's dive in and explore the world of logarithmic equations!
Understanding Logarithms
Before we delve into solving the equation log r + log 4 = 2, it's essential to have a solid grasp of what logarithms are and their fundamental properties. A logarithm is essentially the inverse operation to exponentiation. In simple terms, if we have an exponential equation like b^x = y, the corresponding logarithmic equation is log_b(y) = x. Here, 'b' is the base of the logarithm, 'y' is the argument, and 'x' is the exponent to which 'b' must be raised to obtain 'y'. The most commonly used bases for logarithms are 10 (common logarithm) and 'e' (natural logarithm), denoted as log10(x) or simply log(x), and loge(x) or ln(x), respectively. When the base is not explicitly written, it is generally assumed to be 10. Understanding these basics is crucial because the properties of logarithms allow us to manipulate and simplify complex equations, making them easier to solve. For instance, the product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Similarly, the quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. The power rule states that the logarithm of a number raised to an exponent is equal to the product of the exponent and the logarithm of the number. These properties, along with the definition of a logarithm, are the building blocks for solving logarithmic equations. In the next section, we will apply these concepts to simplify the given equation and isolate the variable 'r'. By understanding the fundamental properties of logarithms, we can approach complex problems with confidence and solve them efficiently. This section serves as a foundation for the subsequent steps, ensuring that you have the necessary knowledge to tackle the equation at hand.
Applying Logarithmic Properties
Now, let's apply the properties of logarithms to simplify the equation log r + log 4 = 2. The first key property we'll use is the product rule of logarithms. This rule states that the sum of the logarithms of two numbers is equal to the logarithm of their product. Mathematically, it can be expressed as: log_b(m) + log_b(n) = log_b(m * n). In our equation, we have log r + log 4, which can be rewritten using the product rule as log (r * 4) or log (4r). So, our equation now becomes log (4r) = 2. The next step is to understand the base of the logarithm. Since the base is not explicitly written, we assume it to be 10 (common logarithm). This means that the equation log (4r) = 2 is equivalent to log10(4r) = 2. To solve for 'r', we need to convert this logarithmic equation into its equivalent exponential form. The relationship between logarithms and exponents is fundamental: if log_b(y) = x, then b^x = y. Applying this to our equation, where b = 10, x = 2, and y = 4r, we get 10^2 = 4r. This conversion is a crucial step in solving logarithmic equations, as it transforms the equation into a more manageable algebraic form. By understanding and applying the product rule and the relationship between logarithms and exponents, we have successfully simplified the original equation and set the stage for solving for 'r'. In the following section, we will continue with the simplification process and isolate 'r' to find its value. This step-by-step approach ensures that the solution is clear and easy to follow, making the process of solving logarithmic equations less daunting.
Solving for r
Having transformed the logarithmic equation into its exponential form, we now have 10^2 = 4r. The next step is to simplify this equation and isolate 'r'. First, we evaluate 10^2, which is simply 10 multiplied by itself, resulting in 100. So, the equation becomes 100 = 4r. To isolate 'r', we need to divide both sides of the equation by 4. This is a fundamental algebraic operation that maintains the equality of the equation while moving us closer to the solution. Dividing both sides by 4, we get 100 / 4 = 4r / 4. Simplifying this, we find that 25 = r. Therefore, the solution to the equation log r + log 4 = 2 is r = 25. It's always a good practice to verify the solution by substituting it back into the original equation. This ensures that our answer is correct and that we haven't made any errors in the process. Substituting r = 25 into the original equation, we get log 25 + log 4 = 2. Using the product rule, we can rewrite this as log (25 * 4) = 2, which simplifies to log 100 = 2. Since log 100 (base 10) is indeed 2, our solution r = 25 is correct. This verification step is crucial in mathematics, as it provides confidence in the answer and reinforces the understanding of the concepts involved. By carefully following the steps, applying the properties of logarithms, and verifying the solution, we have successfully solved the equation. The process demonstrates the importance of understanding the underlying principles and applying them systematically to arrive at the correct answer. In the final section, we will summarize the steps and provide some additional insights into solving logarithmic equations.
Conclusion
In summary, solving the logarithmic equation log r + log 4 = 2 involves several key steps. First, we applied the product rule of logarithms to combine the terms on the left side of the equation, resulting in log (4r) = 2. Next, we converted the logarithmic equation into its equivalent exponential form, using the definition of a logarithm. Since the base of the logarithm was not explicitly written, we assumed it to be 10, leading to the equation 10^2 = 4r. We then simplified the equation by evaluating 10^2, which gave us 100 = 4r. To isolate 'r', we divided both sides of the equation by 4, resulting in r = 25. Finally, we verified our solution by substituting r = 25 back into the original equation, confirming that it satisfies the equation. This step-by-step process highlights the importance of understanding and applying the properties of logarithms, as well as the relationship between logarithms and exponents. Solving logarithmic equations is a fundamental skill in mathematics, and this guide provides a clear and concise explanation of the process. By mastering these techniques, you can confidently tackle similar problems and apply them in various fields that require logarithmic calculations. Remember to always verify your solutions to ensure accuracy and to reinforce your understanding of the concepts. Logarithmic equations are not as daunting as they may seem at first glance. With a solid understanding of the properties of logarithms and a systematic approach, you can solve them effectively. This article serves as a valuable resource for students and professionals alike, providing a comprehensive guide to solving logarithmic equations and building a strong foundation in mathematics.
r = 25
