Solving Log Base 6 Of (-3w + 3) = Log Base 6 Of (-7w - 1) For W

Introduction

In this comprehensive guide, we delve into the intricacies of solving logarithmic equations, focusing specifically on the equation ${\log _6(-3 w+3)=\log _6(-7 w-1)}$. Logarithmic equations often present a unique challenge, requiring a solid understanding of logarithmic properties and algebraic manipulation. Our aim is to provide a clear, step-by-step solution, ensuring that you not only arrive at the correct answer but also grasp the underlying concepts. This article is tailored for students, educators, and anyone with an interest in mathematics, offering an in-depth exploration of the techniques involved in solving such equations. By the end of this guide, you will be well-equipped to tackle similar problems with confidence and precision. We will start by laying the groundwork, reviewing the fundamental properties of logarithms and their role in equation solving. Then, we will proceed to solve the given equation systematically, explaining each step in detail. Finally, we will discuss the importance of verifying solutions in the context of logarithmic equations, as extraneous solutions can sometimes arise. So, let's embark on this mathematical journey and unravel the solution to ${\log _6(-3 w+3)=\log _6(-7 w-1)}$.

Understanding Logarithmic Equations

Before diving into the solution, it's crucial to understand the basics of logarithmic equations. A logarithmic equation is an equation that involves logarithms of expressions containing a variable. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. In simpler terms, if we have ${\log_b a = c}$, it means that ${b^c = a}$, where 'b' is the base, 'a' is the argument, and 'c' is the logarithm. The most common types of logarithms are the common logarithm (base 10), denoted as ${\log_{10}}$ or simply ${\log}$, and the natural logarithm (base e), denoted as ${\log_e}$ or ${\ln}$. Understanding these fundamentals is essential for solving logarithmic equations effectively. One of the key properties we will use in solving our equation is the property that if ${\log_b x = \log_b y}$, then ${x = y}$, provided that x and y are positive and b is a positive number not equal to 1. This property allows us to eliminate the logarithms and work with the arguments directly. However, it's important to remember that the arguments of logarithms must be positive, as the logarithm of a non-positive number is undefined. This constraint will play a crucial role in verifying our solution later. Furthermore, understanding the domain of logarithmic functions is critical. The domain of the function ${f(x) = \log_b x}$ is all positive real numbers, meaning x must be greater than 0. This restriction stems from the definition of logarithms and the fact that exponential functions (which are the inverse of logarithmic functions) always produce positive values. As we proceed to solve the equation, we will keep these principles in mind to ensure the validity of our solution.

Step-by-Step Solution of ${\log _6(-3 w+3)=\log _6(-7 w-1)}$

Now, let's tackle the equation ${\log _6(-3 w+3)=\log _6(-7 w-1)}$ step by step. Our primary goal is to isolate the variable w and determine its value.

Step 1: Apply the Property of Logarithms

Since we have logarithms with the same base (base 6) on both sides of the equation, we can apply the property that if ${\log_b x = \log_b y}$, then ${x = y}$. This allows us to eliminate the logarithms and equate the arguments:

$$-3w + 3 = -7w - 1$$

Step 2: Isolate the Variable w

To solve for w, we need to isolate it on one side of the equation. We can start by adding 7w to both sides:

$$-3w + 7w + 3 = -7w + 7w - 1$$

$$4w + 3 = -1$$

Next, subtract 3 from both sides:

$$4w + 3 - 3 = -1 - 3$$

$$4w = -4$$

Step 3: Solve for w

Finally, divide both sides by 4 to find the value of w:

$$\frac{4w}{4} = \frac{-4}{4}$$

$$w = -1$$

Therefore, the solution to the equation is w = -1. However, we must now verify this solution to ensure it is valid in the original logarithmic equation. This verification step is crucial because logarithmic equations can sometimes produce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation.

Verifying the Solution

Verifying the solution is a critical step in solving logarithmic equations. This is because the arguments of logarithms must be positive. If we plug our solution into the original equation and find that any of the arguments are non-positive, then the solution is extraneous and must be discarded. Let's verify our solution w = -1 in the original equation:

$$\log _6(-3 w+3)=\log _6(-7 w-1)$$

Substitute w = -1 into the arguments:

Argument 1: -3w + 3 = -3(-1) + 3 = 3 + 3 = 6

Argument 2: -7w - 1 = -7(-1) - 1 = 7 - 1 = 6

Both arguments are positive (6 > 0), which means our solution w = -1 is valid. If either of the arguments had been zero or negative, we would have had to reject the solution. This verification process underscores the importance of considering the domain of logarithmic functions when solving logarithmic equations. The domain restriction that the argument must be positive is a key factor in determining the validity of solutions. In this case, since both arguments are positive when w = -1, we can confidently conclude that w = -1 is the correct solution to the equation.

Common Mistakes and How to Avoid Them

When solving logarithmic equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls and understanding how to avoid them is essential for success in this area of mathematics.

1. Forgetting to Verify Solutions: As we've emphasized, verifying solutions is crucial in logarithmic equations. The most common mistake is to find a solution and assume it's correct without checking if it makes the arguments of the logarithms positive. Always substitute your solution back into the original equation and ensure that all logarithmic arguments are greater than zero.

2. Incorrectly Applying Logarithmic Properties: Logarithmic properties are powerful tools, but they must be applied correctly. For example, the property ${\log_b x + \log_b y = \log_b(xy)}$ is valid only when the logarithms have the same base. Misapplying properties can lead to incorrect simplifications and ultimately wrong solutions. It's important to have a solid understanding of each property and its conditions for use.

3. Ignoring the Domain of Logarithmic Functions: The domain of a logarithmic function is restricted to positive numbers. This means that the argument of a logarithm must always be greater than zero. Failing to consider this restriction can lead to extraneous solutions. Before solving a logarithmic equation, it can be helpful to identify the domain restrictions imposed by the logarithms involved.

4. Algebraic Errors: Like any equation-solving process, algebraic errors can occur when manipulating logarithmic equations. These errors can range from simple arithmetic mistakes to incorrect distribution or combining of terms. To minimize algebraic errors, it's important to work neatly, show all steps, and double-check your work.

5. Confusing Logarithmic and Exponential Forms: Logarithmic and exponential forms are inverses of each other, but they are distinct concepts. Confusing the two can lead to errors in rewriting equations or applying properties. Remember that ${\log_b a = c}$ is equivalent to ${b^c = a}$. Understanding this relationship is crucial for converting between the two forms and solving equations effectively. By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving logarithmic equations.

Conclusion

In conclusion, solving the logarithmic equation ${\log _6(-3 w+3)=\log _6(-7 w-1)}$ involves a series of steps that highlight the importance of understanding logarithmic properties, algebraic manipulation, and solution verification. We began by recognizing the fundamental property that if logarithms with the same base are equal, then their arguments must also be equal. This allowed us to transform the logarithmic equation into a simple algebraic equation. We then systematically isolated the variable w through addition, subtraction, and division, arriving at the solution w = -1. However, the journey didn't end there. We emphasized the crucial step of verifying the solution by substituting it back into the original equation. This verification process is essential to ensure that the arguments of the logarithms remain positive, adhering to the domain restrictions of logarithmic functions. By verifying our solution, we confirmed that w = -1 is indeed a valid solution. Throughout this guide, we also highlighted common mistakes that students often make when solving logarithmic equations, such as forgetting to verify solutions, misapplying logarithmic properties, ignoring the domain of logarithmic functions, making algebraic errors, and confusing logarithmic and exponential forms. By being aware of these pitfalls and practicing careful, step-by-step problem-solving, you can enhance your ability to solve logarithmic equations accurately and confidently. Logarithmic equations are a fascinating area of mathematics with applications in various fields, including science, engineering, and finance. Mastering the techniques for solving them not only strengthens your mathematical skills but also opens doors to understanding more complex concepts. We hope this guide has provided you with a clear and comprehensive understanding of how to solve the equation ${\log _6(-3 w+3)=\log _6(-7 w-1)}$ and has equipped you with the knowledge to tackle similar problems with ease.