Solving Logarithmic Equations: A Step-by-Step Guide

Introduction to Logarithmic Equations

Logarithmic equations might seem intimidating at first, but with a clear understanding of the underlying principles, they can be solved systematically. In this guide, we'll tackle the equation log(x+3) = log(5x-5). Our goal is to find the value(s) of x that satisfy this equation. We'll break down the process into manageable steps, ensuring you grasp each concept along the way. Mastering these techniques is crucial for anyone studying mathematics, engineering, or any field that involves quantitative analysis.

When approaching logarithmic equations, it's essential to remember the basic properties of logarithms. A logarithm is essentially the inverse operation of exponentiation. For example, if we have log_b(a) = c, it means that b^c = a. This relationship is fundamental in manipulating and solving logarithmic expressions. Another important concept is the domain of logarithmic functions. The argument of a logarithm (the value inside the logarithm) must always be positive. This constraint often leads to restrictions on the possible values of x in our equations. Therefore, always check your solutions against the original equation to ensure they are valid.

Furthermore, understanding the properties of logarithms, such as the product rule, quotient rule, and power rule, can greatly simplify complex logarithmic expressions. The product rule states that log_b(mn) = log_b(m) + log_b(n), the quotient rule states that log_b(m/n) = log_b(m) - log_b(n), and the power rule states that log_b(m^p) = p * log_b(m). While these rules might not be directly applicable to the simple equation we are solving today, they become invaluable when dealing with more complex logarithmic equations involving multiple terms and exponents. Remember, practice is key to mastering these concepts. Work through various examples and exercises to build your confidence and intuition in solving logarithmic equations. With consistent effort, you'll find that these problems become less daunting and more manageable.

Step 1: Understanding the Equation

The given equation is log(x+3) = log(5x-5). Notice that we have a logarithm on both sides of the equation. A crucial aspect here is that both logarithms have the same base. Although the base isn't explicitly written, it's implied to be base 10 (common logarithm). If the bases were different, we would need to use the change of base formula to make them the same before proceeding.

Understanding the structure of the equation is the first step towards solving it. We recognize that we have a logarithmic function on both sides, which simplifies our approach significantly. This setup allows us to use a key property: if log_b(a) = log_b(c), then a = c. This property is the cornerstone of solving equations where logarithms with the same base are equated. However, it's paramount to remember the domain restrictions. The arguments of the logarithms, in this case, (x+3) and (5x-5), must both be greater than zero. This is because logarithms are only defined for positive arguments. Ignoring this constraint can lead to extraneous solutions, which are values that satisfy the simplified equation but not the original logarithmic equation.

Before diving into the algebraic manipulations, it's beneficial to perform a quick mental check or estimation. This can help you anticipate the potential range of solutions and avoid obvious errors. For instance, if we intuitively guess that x is a small negative number, we can quickly see that (x+3) might still be positive, but (5x-5) would definitely be negative, violating the domain restriction. This kind of preliminary thinking can save time and prevent mistakes. Furthermore, grasping the underlying concepts of logarithms, such as their relationship to exponential functions and their properties, is essential for tackling more complex logarithmic problems. Therefore, ensure you have a solid understanding of these fundamentals before moving on to more advanced topics. Remember, a strong foundation is crucial for success in mathematics.

Step 2: Equating the Arguments

Since the logarithms on both sides of the equation have the same base, we can equate their arguments: x + 3 = 5x - 5. This step is justified by the fundamental property of logarithms that states if log_b(a) = log_b(c), then a = c, provided that a and c are positive. This simplification transforms the logarithmic equation into a simple algebraic equation, making it easier to solve for x.

Equating the arguments is a crucial step that allows us to eliminate the logarithms and work with a more manageable linear equation. However, it's important to emphasize that this step is only valid when the bases of the logarithms are the same. If the bases were different, we would need to apply the change of base formula to make them identical before equating the arguments. The change of base formula is given by log_b(a) = log_c(a) / log_c(b), where c is the new base. Understanding and applying this formula correctly is essential when dealing with logarithms of different bases. Furthermore, it's worth noting that equating the arguments implicitly assumes that both arguments are positive. This assumption must be explicitly checked later to ensure that the solutions obtained are valid. Failing to verify this condition can lead to incorrect answers.

Before proceeding further, take a moment to reflect on what we have achieved. By equating the arguments, we have successfully transformed a potentially complex logarithmic equation into a simple linear equation that can be solved using basic algebraic techniques. This is a testament to the power of understanding the properties of logarithms and applying them strategically. Remember, mathematics is not just about memorizing formulas, but about understanding the underlying principles and using them to simplify problems. With a clear understanding of these principles, you'll be well-equipped to tackle even more challenging logarithmic equations in the future. Keep practicing and reinforcing your knowledge, and you'll find that mathematics becomes increasingly intuitive and enjoyable.

Step 3: Solving for x

Now we solve the linear equation x + 3 = 5x - 5. Let's isolate x:

1. Add 5 to both sides: x + 8 = 5x
2. Subtract x from both sides: 8 = 4x
3. Divide both sides by 4: x = 2

Therefore, the solution to the equation x + 3 = 5x - 5 is x = 2. This value of x is a potential solution to the original logarithmic equation, but it's crucial to verify whether it satisfies the domain restrictions.

Solving for x involves a series of algebraic manipulations aimed at isolating the variable on one side of the equation. Each step must be performed carefully and accurately to avoid errors. Adding or subtracting the same value from both sides maintains the equality, as does multiplying or dividing both sides by the same non-zero value. The key is to apply these operations strategically to gradually simplify the equation until x is isolated. In this case, we first added 5 to both sides to eliminate the constant term on the right side, then subtracted x from both sides to collect the x terms on one side, and finally divided by 4 to solve for x. While the steps themselves are relatively straightforward, it's essential to pay attention to detail and avoid common algebraic mistakes, such as incorrect sign changes or arithmetic errors.

Once we have found a potential solution for x, it's tempting to assume that we have solved the problem. However, in the context of logarithmic equations, it's crucial to remember that not all solutions to the simplified equation are necessarily solutions to the original equation. This is because logarithmic functions have domain restrictions, meaning they are only defined for certain values of the argument. Therefore, we must always check our solutions against the original equation to ensure they satisfy these domain restrictions. This verification step is often overlooked but is absolutely essential for obtaining correct answers.

Step 4: Checking the Solution

We must check if x = 2 is a valid solution by plugging it back into the original equation log(x+3) = log(5x-5).

• Left side: log(2 + 3) = log(5)
• Right side: log(5(2) - 5) = log(10 - 5) = log(5)

Since both sides are equal, x = 2 is indeed a solution. But, we also need to check if the arguments of the logarithms are positive for this value of x.

• x + 3 = 2 + 3 = 5 > 0
• 5x - 5 = 5(2) - 5 = 5 > 0

Both arguments are positive, so x = 2 is a valid solution.

Checking the solution is a critical step in solving any equation, but it's particularly important for logarithmic and radical equations. This step ensures that the solution we obtained algebraically is also a valid solution in the context of the original problem. In the case of logarithmic equations, we need to verify two conditions: first, that the solution satisfies the original equation when plugged in, and second, that the arguments of all logarithms are positive for that solution. If either of these conditions is not met, then the solution is extraneous and must be discarded.

In this example, we found that x = 2 satisfies both conditions. When we plug it back into the original equation, we find that both sides are equal to log(5), indicating that the solution is consistent with the equation. Furthermore, we checked that both x + 3 and 5x - 5 are positive when x = 2, which ensures that the logarithms are defined for this value of x. Since both conditions are met, we can confidently conclude that x = 2 is a valid solution to the logarithmic equation.

It's worth noting that sometimes, when solving logarithmic equations, we may encounter situations where the algebraic solution does not satisfy the domain restrictions. In such cases, we would say that the equation has no solution, or that the solution set is empty. Therefore, it's essential to be vigilant in checking the solution and to understand the implications of domain restrictions in logarithmic equations. This attention to detail will help you avoid mistakes and ensure that you arrive at the correct answer.

Conclusion

Therefore, the solution to the logarithmic equation log(x+3) = log(5x-5) is x = 2. By understanding the properties of logarithms and following a systematic approach, we successfully solved the equation. Remember to always check your solutions to ensure they are valid.

Solving logarithmic equations requires a solid understanding of logarithmic properties and a careful, step-by-step approach. We began by understanding the equation and recognizing that the logarithms on both sides had the same base. This allowed us to equate the arguments, transforming the logarithmic equation into a simpler algebraic equation. We then solved for x using basic algebraic manipulations and found a potential solution. However, we didn't stop there. We recognized the importance of checking our solution to ensure it was valid and satisfied the domain restrictions of the original equation. By plugging the solution back into the original equation and verifying that the arguments of the logarithms were positive, we confirmed that x = 2 was indeed a valid solution.

This example illustrates the key steps involved in solving logarithmic equations: understanding the equation, applying logarithmic properties to simplify it, solving the resulting algebraic equation, and, most importantly, checking the solution. While the specific steps may vary depending on the complexity of the equation, the underlying principles remain the same. By mastering these principles and practicing regularly, you can develop the skills and confidence needed to tackle a wide range of logarithmic equations. Remember, mathematics is a journey of discovery, and each problem you solve brings you one step closer to a deeper understanding of the subject.

In conclusion, solving logarithmic equations is a fundamental skill in mathematics with applications in various fields. By following a systematic approach and understanding the underlying principles, you can successfully solve these equations and gain a deeper appreciation for the power and elegance of mathematics. For further information on logarithms visit Khan Academy.