Solving Logarithmic Equations Find The Value Of X

Introduction

In the realm of mathematics, solving logarithmic equations is a fundamental skill. In this comprehensive guide, we will delve into the intricacies of solving the equation $\log _2(6 x-8)-\log _2 8=1$. Our primary goal is to determine the precise value of $x$ that satisfies this equation. Before we embark on the step-by-step solution, let's lay a solid foundation by understanding the core concepts of logarithms and their properties. This will equip you with the necessary tools to confidently tackle not only this specific equation but also a wide array of logarithmic problems you may encounter in your mathematical journey. We will explore the definition of logarithms, the various properties that govern their behavior, and how these properties can be strategically applied to simplify and solve equations. Furthermore, we will emphasize the importance of checking for extraneous solutions, a crucial step in ensuring the validity of our final answer. By the end of this guide, you will have a clear and concise understanding of how to approach logarithmic equations and arrive at accurate solutions.

Understanding Logarithms

Before diving into the equation, it's crucial to understand the basics of logarithms. A logarithm is essentially the inverse operation of exponentiation. In simpler terms, if we have an equation like $b^y = x$, the logarithm of $x$ to the base $b$ is $y$. This is written as $\log_b x = y$. The base, $b$, is a crucial part of the logarithm, and it tells us what number is being raised to a power. In our given equation, the base is 2. To solidify your understanding, let's consider a few examples. The expression $\log_2 8$ asks the question, "To what power must we raise 2 to get 8?" The answer is 3, since $2^3 = 8$. Similarly, $\log_{10} 100$ asks, "To what power must we raise 10 to get 100?" The answer is 2, since $10^2 = 100$. These examples illustrate the fundamental relationship between logarithms and exponents. Grasping this relationship is key to effectively manipulating and solving logarithmic equations. Now that we have a firm grasp of the basic definition of logarithms, let's explore the properties that govern their behavior. These properties will be instrumental in simplifying our equation and isolating the variable $x$. Understanding the properties of logarithms not only allows us to solve equations more efficiently but also provides a deeper appreciation for the elegance and power of this mathematical concept.

Key Properties of Logarithms

Several key properties of logarithms are essential for solving equations. The most relevant property for our problem is the quotient rule, which states that the logarithm of a quotient is equal to the difference of the logarithms. Mathematically, this is expressed as $\log_b(x/y) = \log_b x - \log_b y$. This property allows us to combine logarithmic terms that are being subtracted. Another crucial property is the power rule, which states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. This is written as $\log_b(x^p) = p \log_b x$. This property is useful for simplifying expressions where the argument of the logarithm is raised to a power. Additionally, we have the product rule, which states that the logarithm of a product is equal to the sum of the logarithms: $\log_b(xy) = \log_b x + \log_b y$. Understanding and applying these properties correctly is crucial for simplifying logarithmic expressions and solving equations effectively. These properties are not just abstract rules; they are powerful tools that allow us to manipulate logarithmic expressions in a way that makes them easier to work with. In the context of our equation, the quotient rule will play a pivotal role in combining the logarithmic terms and ultimately isolating the variable $x$. Now that we have reviewed the fundamental properties, let's apply them to solve our equation step-by-step.

Solving the Equation Step-by-Step

Now, let's solve the equation $\log _2(6 x-8)-\log _2 8=1$ step-by-step. Our primary goal is to isolate $x$ and determine its value. The first step involves using the quotient rule of logarithms to combine the two logarithmic terms on the left side of the equation. Recall that the quotient rule states that $\log_b(x/y) = \log_b x - \log_b y$. Applying this rule to our equation, we can rewrite the left side as a single logarithm: $\log_2((6x-8)/8) = 1$. This step is crucial because it simplifies the equation by reducing the number of logarithmic terms. Now that we have a single logarithm on one side, we can proceed to the next step, which involves converting the logarithmic equation into its equivalent exponential form. Remember that a logarithm is the inverse of an exponent, so we can rewrite the equation in exponential form by using the definition of a logarithm. This will allow us to eliminate the logarithm altogether and work with a simpler algebraic equation. Once we have the equation in exponential form, we can proceed with standard algebraic techniques to isolate $x$ and find its value. Each step in this process is carefully designed to bring us closer to the solution while maintaining the equality of the equation. Let's move on to the next step and see how we can transform our logarithmic equation into an exponential one.

Step 1: Apply the Quotient Rule

Applying the quotient rule, we combine the logarithms: $\log _2\left(\frac{6 x-8}{8}\right)=1$. This simplifies the equation and makes it easier to work with. The quotient rule is a powerful tool in simplifying logarithmic expressions, and its application here is a crucial step in our solution process. By combining the two logarithms into a single one, we have effectively reduced the complexity of the equation. This allows us to move forward with the solution process with a clearer path ahead. The quotient rule is not just a mathematical trick; it is a fundamental property of logarithms that reflects the relationship between logarithmic and exponential functions. Understanding and applying this rule correctly is essential for solving logarithmic equations efficiently and accurately. Now that we have simplified the equation using the quotient rule, the next step is to convert it from logarithmic form to exponential form. This will allow us to eliminate the logarithm and work with a more familiar algebraic equation. Let's proceed to the next step and see how this transformation unfolds.

Step 2: Convert to Exponential Form

To eliminate the logarithm, we convert the equation to exponential form. The equation $\log _2\left(\frac{6 x-8}{8}\right)=1$ in exponential form is $2^1 = \frac{6x-8}{8}$. This step is crucial as it removes the logarithm, allowing us to solve for $x$ using basic algebra. Converting from logarithmic form to exponential form is a fundamental skill in solving logarithmic equations. It relies on the inverse relationship between logarithms and exponents. Understanding this relationship is key to making this conversion correctly. In our case, the base of the logarithm is 2, and the exponent is 1, so we raise 2 to the power of 1. This eliminates the logarithm and gives us a simple algebraic equation that we can solve for $x$. This step is like unlocking a door that was previously blocked by the logarithm. Once we have the equation in exponential form, we can use standard algebraic techniques to isolate $x$ and find its value. Let's move on to the next step and see how we can manipulate this equation to solve for our unknown variable.

Step 3: Simplify and Solve for x

Simplifying the equation $2^1 = \frac{6x-8}{8}$, we get $2 = \frac{6x-8}{8}$. Multiplying both sides by 8 gives us $16 = 6x - 8$. Adding 8 to both sides yields $24 = 6x$. Finally, dividing both sides by 6, we find that $x = 4$. This step involves basic algebraic manipulation to isolate $x$. Each operation we perform on the equation must be done on both sides to maintain equality. This is a fundamental principle of algebra that ensures we are transforming the equation without changing its solution. The process of simplifying and solving for $x$ involves a series of steps, each building upon the previous one. We start by multiplying both sides by 8 to eliminate the fraction. Then, we add 8 to both sides to isolate the term containing $x$. Finally, we divide both sides by 6 to solve for $x$. This step-by-step approach allows us to systematically unravel the equation and arrive at the solution. Now that we have found a potential value for $x$, it is crucial to verify that this value is indeed a valid solution. This is because logarithmic equations can sometimes have extraneous solutions, which are values that satisfy the transformed equation but not the original equation. Let's move on to the next step and see how we can check our solution.

Step 4: Check for Extraneous Solutions

It's essential to check our solution in the original equation. Substituting $x = 4$ into the original equation, we get $\log _2(6(4)-8)-\log _2 8=1$. This simplifies to $\log _2(24-8)-\log _2 8=1$, then $\log _2(16)-\log _2 8=1$. Since $\log _2(16) = 4$ and $\log _2(8) = 3$, the equation becomes $4 - 3 = 1$, which is true. Therefore, $x = 4$ is a valid solution. Checking for extraneous solutions is a crucial step in solving logarithmic equations. This is because the domain of a logarithmic function is restricted to positive numbers. If we substitute a value for $x$ that results in a negative argument inside the logarithm, the logarithm is undefined, and the solution is extraneous. In our case, we need to make sure that both $6x - 8$ and 8 are positive when $x = 4$. Since $6(4) - 8 = 16$ and 8 is positive, our solution is valid. This step ensures that we have not introduced any errors during the solution process and that the value we have found truly satisfies the original equation. Now that we have verified our solution, we can confidently state the final answer.

Final Answer

The value of $x$ that satisfies the equation $\log _2(6 x-8)-\log _2 8=1$ is $4$.

Conclusion

In this comprehensive guide, we have meticulously walked through the process of solving the logarithmic equation $\log _2(6 x-8)-\log _2 8=1$. We began by establishing a solid understanding of logarithms, their fundamental properties, and their relationship to exponential functions. We then strategically applied these properties, particularly the quotient rule, to simplify the equation and isolate the variable $x$. We converted the logarithmic equation into its equivalent exponential form, which allowed us to eliminate the logarithm and work with a simpler algebraic equation. Through a series of algebraic manipulations, we successfully solved for $x$ and arrived at the potential solution $x = 4$. However, we did not stop there. Recognizing the importance of verifying solutions in logarithmic equations, we meticulously checked our solution in the original equation. This crucial step ensured that we had not introduced any extraneous solutions and that our answer was indeed valid. By substituting $x = 4$ back into the original equation, we confirmed that it satisfied the equation, thereby validating our solution. This process highlights the importance of not only finding a solution but also verifying its correctness, especially in the context of logarithmic equations. The final answer, $x = 4$, represents the precise value that satisfies the given equation. By mastering the techniques and concepts presented in this guide, you are now well-equipped to tackle a wide range of logarithmic equations with confidence and accuracy. Remember to always start with a clear understanding of the properties of logarithms, apply them strategically to simplify the equation, and always check your solutions to ensure their validity. With practice and perseverance, you will become proficient in solving logarithmic equations and further enhance your mathematical skills.