Solving Logarithmic Equations A Step By Step Guide To Solve Log₅(x) - Log₅(8x - 1) = 0
Introduction
In this article, we will delve into solving the logarithmic equation log₅(x) - log₅(8x - 1) = 0. Logarithmic equations often appear complex, but with a systematic approach and a solid understanding of logarithmic properties, they can be solved effectively. This comprehensive guide will break down the problem step by step, ensuring you grasp the underlying concepts and techniques. We will start by reviewing the fundamental properties of logarithms, then proceed to simplify the equation, and finally, solve for x. Additionally, we will discuss the importance of verifying solutions in the context of logarithmic equations to avoid extraneous roots. By the end of this article, you will not only be able to solve this particular equation but also gain valuable insights into tackling similar logarithmic problems. Whether you're a student seeking to improve your math skills or someone looking to refresh your knowledge, this guide provides a clear and concise method for solving logarithmic equations.
Understanding Logarithmic Properties
Before we dive into the solution, it’s crucial to understand the basic properties of logarithms. Logarithms are the inverse operations of exponentiation, and they follow several key rules that are essential for solving logarithmic equations. Firstly, the logarithmic function logₐ(b) = c can be rewritten in exponential form as aᶜ = b. This fundamental relationship allows us to convert between logarithmic and exponential forms, which is often necessary when simplifying equations. Secondly, we have the logarithm of a quotient rule, which states that logₐ(x) - logₐ(y) = logₐ(x/y). This property is particularly useful in our given equation, as it involves the difference of two logarithms with the same base. Another important property is the logarithm of a product rule, which states that logₐ(x) + logₐ(y) = logₐ(xy). While not directly applicable in this problem, it's a valuable concept to keep in mind for other logarithmic equations. Additionally, we need to remember that the domain of a logarithmic function is restricted to positive numbers, meaning that the argument of a logarithm (the value inside the parentheses) must be greater than zero. This is a critical consideration when verifying our solutions to ensure they are valid. Understanding these properties forms the foundation for solving logarithmic equations effectively. As we proceed with the solution, we will apply these rules to simplify the equation and isolate the variable x.
Step-by-Step Solution
To solve the equation log₅(x) - log₅(8x - 1) = 0, we will follow a systematic approach, applying the properties of logarithms to simplify and isolate the variable x. The first step involves using the logarithm of a quotient property, which states that logₐ(x) - logₐ(y) = logₐ(x/y). Applying this to our equation, we can rewrite log₅(x) - log₅(8x - 1) as log₅(x / (8x - 1)). So, our equation now becomes log₅(x / (8x - 1)) = 0. The next step is to convert this logarithmic equation into its equivalent exponential form. Recall that logₐ(b) = c can be rewritten as aᶜ = b. Applying this to our equation, we get 5⁰ = x / (8x - 1). Since any non-zero number raised to the power of 0 is 1, we have 1 = x / (8x - 1). Now, we need to solve this algebraic equation for x. To do this, we multiply both sides of the equation by (8x - 1), which gives us 8x - 1 = x. Next, we subtract x from both sides to get 7x - 1 = 0. Adding 1 to both sides, we have 7x = 1. Finally, we divide both sides by 7 to solve for x, which gives us x = 1/7. This value of x is a potential solution, but it is crucial to verify it in the original equation to ensure it does not lead to any undefined logarithms.
Verifying the Solution
After finding a potential solution, it is crucial to verify the solution in the original equation. This step is particularly important when dealing with logarithmic equations because logarithms are only defined for positive arguments. Substituting the value x = 1/7 back into the original equation log₅(x) - log₅(8x - 1) = 0, we need to check if both logarithmic terms are defined and if the equation holds true. First, let's check the argument of the first logarithm, log₅(x). Substituting x = 1/7, we get log₅(1/7), which is defined since 1/7 is a positive number. Next, we check the argument of the second logarithm, log₅(8x - 1). Substituting x = 1/7, we get 8(1/7) - 1 = 8/7 - 1 = 8/7 - 7/7 = 1/7. So, we have log₅(1/7), which is also defined. Now, let's substitute x = 1/7 into the original equation: log₅(1/7) - log₅(8(1/7) - 1) = log₅(1/7) - log₅(1/7). Since subtracting a value from itself equals zero, we have 0 = 0, which confirms that x = 1/7 is indeed a valid solution. This verification step ensures that our solution is not an extraneous root, which can sometimes arise when solving logarithmic or radical equations. By carefully checking the domain restrictions, we can be confident in the accuracy of our solution.
Common Mistakes to Avoid
When solving logarithmic equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct solution. One frequent mistake is forgetting to verify the solutions. As discussed earlier, logarithms are only defined for positive arguments, so it's essential to check if the solutions obtained satisfy the domain restrictions of the logarithmic functions involved. Failing to do so can lead to the inclusion of extraneous solutions. Another common error is misapplying the properties of logarithms. For instance, incorrectly applying the quotient rule or the product rule can lead to significant errors in the simplification process. It's crucial to have a solid understanding of these properties and apply them correctly. Another mistake is incorrectly converting between logarithmic and exponential forms. The relationship between logₐ(b) = c and aᶜ = b must be clearly understood to avoid errors in the conversion process. Additionally, students sometimes make algebraic errors when solving the resulting equations after applying logarithmic properties. It’s important to pay close attention to detail and double-check each step to avoid mistakes in algebraic manipulations. Finally, overlooking the base of the logarithm is another potential error. The properties of logarithms depend on having the same base, so it's essential to ensure that the bases are consistent before applying any logarithmic rules. By being mindful of these common mistakes and practicing a systematic approach, you can improve your accuracy and confidence in solving logarithmic equations.
Conclusion
In summary, we have successfully solved the logarithmic equation log₅(x) - log₅(8x - 1) = 0 by applying the properties of logarithms and following a step-by-step approach. We began by understanding the fundamental properties of logarithms, including the quotient rule, which allowed us to simplify the equation into log₅(x / (8x - 1)) = 0. We then converted the logarithmic equation into its exponential form, resulting in 5⁰ = x / (8x - 1), which simplified to 1 = x / (8x - 1). By solving this algebraic equation, we found a potential solution of x = 1/7. Crucially, we verified this solution by substituting it back into the original equation and confirming that it satisfied the domain restrictions of the logarithmic functions and the equation itself. This verification step is essential to avoid extraneous solutions and ensures the accuracy of our result. We also discussed common mistakes to avoid when solving logarithmic equations, such as forgetting to verify solutions, misapplying logarithmic properties, making algebraic errors, and overlooking the base of the logarithm. By understanding these pitfalls and practicing a systematic approach, you can enhance your ability to solve logarithmic equations effectively. The techniques and concepts covered in this article provide a solid foundation for tackling a wide range of logarithmic problems. With practice and a clear understanding of the underlying principles, you can confidently solve logarithmic equations and apply these skills in various mathematical contexts.
