Solution To The Logarithmic Equation 2log₉(x) = Log₉(8) + Log₉(x-2)

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In this article, we will delve into the process of solving the logarithmic equation 2 log₉(x) = log₉(8) + log₉(x-2). This type of equation, involving logarithms, often appears in algebra and calculus, and understanding how to solve them is crucial for various mathematical applications. We will explore the properties of logarithms, apply them step-by-step, and discuss the importance of checking for extraneous solutions. The ultimate goal is to find the value(s) of 'x' that satisfy the given equation. This exploration will not only provide the solution but also enhance your understanding of logarithmic functions and their manipulations. Grasping these concepts is essential for more advanced mathematical studies and problem-solving scenarios where logarithmic equations frequently arise. So, let's embark on this journey to unravel the solution to this logarithmic puzzle, step by step, ensuring a clear and comprehensive understanding along the way. Our approach will focus on clarity, precision, and a thorough explanation of each step involved, making it accessible even to those who may find logarithmic equations challenging at first glance.

Understanding Logarithmic Properties

Before diving into the solution, it's essential to understand the fundamental properties of logarithms that will be used throughout the process. Logarithmic properties are the cornerstone of solving logarithmic equations. They allow us to manipulate and simplify expressions, ultimately leading to the isolation of the variable. The key properties include the power rule, the product rule, and the quotient rule. Let's briefly review these to ensure we have a solid foundation for the subsequent steps. The power rule states that logₐ(xⁿ) = n logₐ(x), which allows us to move exponents in front of the logarithm. The product rule states that logₐ(xy) = logₐ(x) + logₐ(y), enabling us to combine the logarithms of products into sums. The quotient rule, conversely, states that logₐ(x/y) = logₐ(x) - logₐ(y), allowing us to separate the logarithms of quotients into differences. Moreover, a crucial understanding is that logarithms are only defined for positive arguments. This implies that when solving logarithmic equations, we must always check our solutions to ensure that they do not result in taking the logarithm of a non-positive number. This verification step is critical in identifying and discarding extraneous solutions, which are values obtained during the solving process that do not actually satisfy the original equation. A clear grasp of these properties and the domain restrictions of logarithms is the bedrock upon which we will build our solution strategy.

Step-by-Step Solution

Let's begin by rewriting the given equation: 2 log₉(x) = log₉(8) + log₉(x-2). Our first step involves applying the power rule of logarithms to the left side of the equation. This rule states that n logₐ(x) = logₐ(xⁿ). Applying this, we can rewrite 2 log₉(x) as log₉(x²). Now our equation becomes: log₉(x²) = log₉(8) + log₉(x-2). Next, we will use the product rule of logarithms, which states that logₐ(x) + logₐ(y) = logₐ(xy). We apply this rule to the right side of the equation, combining the two logarithms into a single one: log₉(x²) = log₉(8(x-2)). Now that we have a single logarithm on each side of the equation with the same base (base 9), we can equate the arguments of the logarithms. This means we can remove the logarithms and set the expressions inside them equal to each other: x² = 8(x-2). This simplifies the equation to a quadratic equation, which we can solve using standard algebraic techniques. Expanding the right side gives us x² = 8x - 16. Rearranging the terms to one side, we get a quadratic equation in standard form: x² - 8x + 16 = 0. This equation is now ready to be solved, and we will proceed to find its roots in the subsequent steps.

Solving the Quadratic Equation

The quadratic equation we obtained is x² - 8x + 16 = 0. We can solve this equation by factoring, completing the square, or using the quadratic formula. In this case, the equation is easily factorable. We are looking for two numbers that multiply to 16 and add up to -8. These numbers are -4 and -4. Therefore, we can factor the quadratic equation as follows: (x - 4)(x - 4) = 0. This can also be written as (x - 4)² = 0. To find the solution(s) for x, we set each factor equal to zero: x - 4 = 0. Solving for x, we find x = 4. In this case, we have a repeated root, meaning that x = 4 is the only solution obtained from solving the quadratic equation. However, it is crucial to remember that we are dealing with a logarithmic equation, and we must check if this solution is valid by plugging it back into the original equation. This step is essential to ensure that the solution does not result in taking the logarithm of a non-positive number, which is undefined. Therefore, the next step is to verify the solution in the context of the original logarithmic equation.

Checking for Extraneous Solutions

Now, we need to check if the solution x = 4 is valid for the original equation: 2 log₉(x) = log₉(8) + log₉(x-2). We substitute x = 4 into the original equation and evaluate both sides to see if they are equal. Substituting x = 4, we get: 2 log₉(4) = log₉(8) + log₉(4-2). Simplifying the equation, we have: 2 log₉(4) = log₉(8) + log₉(2). Now, we can apply the power rule on the left side: log₉(4²) = log₉(16). On the right side, we use the product rule: log₉(8) + log₉(2) = log₉(8 * 2) = log₉(16). So, we have: log₉(16) = log₉(16). Since both sides of the equation are equal, x = 4 is indeed a valid solution. Furthermore, we need to ensure that the arguments of the logarithms in the original equation are positive when x = 4. In the original equation, the arguments are x and x - 2. When x = 4, both arguments are positive (4 > 0 and 4 - 2 = 2 > 0), confirming that x = 4 is a legitimate solution. Therefore, after verifying the solution in the original equation and ensuring the arguments of the logarithms are positive, we can confidently conclude that x = 4 is the solution to the given logarithmic equation.

Final Answer

After carefully solving the logarithmic equation 2 log₉(x) = log₉(8) + log₉(x-2), we have arrived at the solution. We systematically applied the properties of logarithms, transformed the equation into a quadratic equation, solved the quadratic equation, and, most importantly, checked for extraneous solutions. The solution we obtained was x = 4. Upon substituting this value back into the original equation, we verified that it satisfies the equation and does not result in taking the logarithm of a non-positive number. Therefore, the final answer to the equation is x = 4. This process highlights the importance of not only understanding the algebraic manipulations involved in solving equations but also the critical step of verifying solutions in the context of the original equation, especially when dealing with logarithmic and radical equations. This comprehensive approach ensures accuracy and a thorough understanding of the problem-solving process.

Therefore, the solution to the equation 2 log₉(x) = log₉(8) + log₉(x-2) is C. x = 4.