Solving Logarithmic Equations A Step-by-Step Guide
This article will walk you through the process of solving the logarithmic equation logāx + logā3 = logā(x+1). Logarithmic equations can seem intimidating at first, but with a systematic approach and a solid understanding of logarithmic properties, they become quite manageable. This article aims to provide a comprehensive explanation, ensuring clarity and ease of understanding for anyone venturing into the realm of logarithmic problem-solving. This includes understanding the foundational logarithmic properties, employing algebraic manipulations, and arriving at a definitive solution while verifying its validity. The step-by-step guide is designed to demystify the equation and empower readers to tackle similar problems confidently. Logarithmic equations appear frequently in mathematical and scientific contexts, making their mastery essential. We will begin by revisiting core logarithmic principles, then apply them to simplify the given equation, and methodically solve for the unknown variable. By the end of this exploration, you should not only understand the solution to this specific problem but also have acquired a robust strategy for dealing with logarithmic challenges in the future. Dive in and unravel the mysteries of logarithms!
Understanding Logarithmic Properties
Before diving into the solution, it's crucial to understand the fundamental properties of logarithms. Logarithms are essentially the inverse operations of exponentiation. The logarithmic function, denoted as logā(x), asks the question: "To what power must we raise the base 'a' to obtain the value 'x'?" The properties that govern logarithmic operations are vital tools in simplifying and solving logarithmic equations. These properties provide the bedrock upon which logarithmic manipulations are built. Among the most frequently used are the product rule, the quotient rule, and the power rule. The product rule states that the logarithm of a product is the sum of the logarithms of the individual factors, a cornerstone for simplifying expressions involving multiplication within logarithms. Similarly, the quotient rule simplifies the logarithm of a quotient into the difference of logarithms, invaluable for division-related logarithmic simplification. Then, the power rule gracefully handles exponents within logarithms, turning complex powers into simpler multiplications. These properties are not just mathematical abstractions; they are the practical instruments that make solving logarithmic equations a structured and achievable process. By mastering them, one gains the ability to decompose complex logarithmic expressions into manageable components, facilitating the isolation and determination of unknown variables. A thorough grasp of these principles transforms what may initially appear as daunting equations into approachable puzzles, solvable with logical and strategic steps. This fundamental understanding is the key to unlocking the power and elegance of logarithmic mathematics.
The Product Rule
The product rule is a cornerstone of logarithmic operations, elegantly stating that the logarithm of a product is equivalent to the sum of the logarithms of the individual factors. Mathematically, this is represented as logā(mn) = logā(m) + logā(n), where 'a' is the base of the logarithm and 'm' and 'n' are positive numbers. This rule is not just a formula; it's a powerful tool that simplifies complex expressions by transforming multiplication inside a logarithm into addition outside of it. It allows us to break down intricate problems into smaller, more manageable parts, making it a fundamental property for solving equations and simplifying expressions. Understanding the product rule enables us to manipulate logarithmic equations with greater flexibility, converting products into sums and vice versa. This capability is particularly useful in situations where isolating variables or simplifying expressions is crucial. The product rule isn't just a mathematical trick; it is a reflection of the inherent relationship between logarithms and exponents. By mastering this rule, we gain a deeper insight into the nature of logarithms and their applications in various fields, including mathematics, physics, and engineering. It is an indispensable tool in the arsenal of anyone dealing with logarithmic functions and equations, providing a clear and logical pathway to simplification and problem-solving.
Applying the Product Rule to Our Equation
In our specific equation, logāx + logā3 = logā(x+1), we can apply the product rule to the left side of the equation. The product rule states that the sum of logarithms with the same base is equal to the logarithm of the product of their arguments. In this case, we have logāx + logā3, which can be rewritten as logā(x * 3), or simply logā(3x). Applying the product rule transforms the sum of two logarithms into a single logarithm, simplifying the equation and bringing us closer to a solution. This step is crucial because it consolidates the logarithmic terms on the left side, making it easier to compare with the right side of the equation. The transformation from logāx + logā3 to logā(3x) is not merely a manipulation of symbols; it is a fundamental simplification that leverages the core properties of logarithms. By combining the two logarithms into one, we reduce the complexity of the equation and pave the way for further algebraic operations. This application of the product rule demonstrates its practical utility in simplifying logarithmic expressions and setting the stage for solving logarithmic equations. It is a critical step in our journey to find the value of x that satisfies the original equation, highlighting the importance of understanding and applying logarithmic properties correctly.
Solving the Equation
Now that we've applied the product rule, our equation looks like this: logā(3x) = logā(x+1). The next step is to eliminate the logarithms. Since both sides of the equation have logarithms with the same base (base 6), we can equate the arguments of the logarithms. This principle is based on the fact that logarithmic functions are one-to-one, meaning that if the logarithms of two quantities are equal, then the quantities themselves must be equal. This transition from logarithmic equality to algebraic equality is a pivotal moment in solving the equation. It allows us to shift from the realm of logarithms to the more familiar territory of algebraic manipulation. By equating the arguments, we are essentially stripping away the logarithmic veneer and focusing on the underlying algebraic relationship. This step simplifies the equation significantly, making it more accessible to standard algebraic techniques. The ability to equate arguments directly stems from the fundamental properties of logarithmic functions and their inverse relationship with exponential functions. This principle is not just a shortcut; it is a logical consequence of the definition of logarithms. With the arguments equated, we now have a linear equation that can be easily solved using basic algebraic operations. This is a testament to the power of logarithmic properties in transforming complex equations into simpler, solvable forms. From this point forward, the solution path is clear, involving straightforward algebraic steps to isolate the variable and determine its value.
Equating the Arguments
Equating the arguments means setting the expressions inside the logarithms equal to each other. In our case, this means setting 3x equal to x+1. This step is valid because the logarithmic function is one-to-one; if logā(m) = logā(n), then m = n. This property is fundamental to solving logarithmic equations, allowing us to move from a logarithmic equation to a simpler algebraic equation. By equating the arguments, we bypass the complexities of logarithmic functions and focus on the underlying relationship between the variables. This transformation is not arbitrary; it is a direct application of the inherent properties of logarithms. The one-to-one nature of logarithmic functions ensures that this step preserves the equality of the equation. Equating arguments is a crucial technique in solving logarithmic equations, providing a clear and direct path to finding the solution. It is a testament to the elegance and power of mathematical principles, where a seemingly complex equation can be reduced to a simpler form through the application of fundamental properties. This step is not just a mathematical trick; it is a logical deduction based on the definition and characteristics of logarithms. Once the arguments are equated, we are left with an algebraic equation that can be solved using standard techniques, making the problem much more approachable.
Solving for x
Now we have the equation 3x = x + 1. To solve for x, we need to isolate x on one side of the equation. This involves basic algebraic manipulations, which are the bread and butter of equation solving. First, we subtract x from both sides of the equation. This step maintains the balance of the equation while bringing all the x terms to one side. The result is 2x = 1. Next, to isolate x completely, we divide both sides of the equation by 2. This gives us x = 1/2. These algebraic manipulations are not just rote steps; they are logical operations that follow the rules of equality. Subtracting the same quantity from both sides and dividing both sides by the same non-zero quantity are fundamental operations that preserve the balance of the equation. The goal is to systematically isolate the variable, and each step is designed to bring us closer to that goal. Solving for x involves a careful and methodical application of these algebraic principles. The process is like peeling away layers to reveal the value of x at the core. The solution x = 1/2 is the value that satisfies the algebraic equation, but it is crucial to remember that we started with a logarithmic equation. Therefore, we need to check this solution in the original logarithmic equation to ensure it is valid.
Checking the Solution
After finding a potential solution, it's imperative to check it in the original equation. This step is crucial because logarithmic functions have domain restrictions. The argument of a logarithm must be positive, meaning we can only take the logarithm of positive numbers. This constraint is a fundamental aspect of logarithmic functions and must be considered when solving logarithmic equations. Checking the solution involves substituting the value we found for x back into the original equation and verifying that it satisfies the equation without violating any domain restrictions. This process is not just a formality; it is a safeguard against extraneous solutions. Extraneous solutions are values that satisfy the simplified algebraic equation but do not satisfy the original logarithmic equation due to domain restrictions. These solutions can arise from the algebraic manipulations we perform during the solving process. Therefore, checking the solution is a critical step in ensuring that the answer we obtain is valid and meaningful. It is a practice that reflects the rigor of mathematical problem-solving, where every solution must be verified against the initial conditions and constraints. By checking our solution, we ensure that we have not introduced any errors or extraneous roots and that our final answer is correct and complete. This step provides confidence in the solution and demonstrates a thorough understanding of logarithmic functions.
Substituting x = 1/2 into the Original Equation
Let's substitute x = 1/2 into the original equation: logāx + logā3 = logā(x+1). Replacing x with 1/2, we get logā(1/2) + logā3 = logā(1/2 + 1). Now we need to simplify and verify if both sides of the equation are equal. This substitution is a direct application of our solution back into the initial equation, the litmus test for its validity. The act of substitution is not just a mechanical step; it is a crucial verification process that ensures our solution respects the constraints of the original problem. It is a tangible way to confirm that the value we found for x truly satisfies the given conditions. By substituting and simplifying, we are essentially retracing our steps, ensuring that each transformation and manipulation we performed was valid and did not introduce any errors. This process of verification is a hallmark of rigorous mathematical practice, where no solution is accepted without thorough scrutiny. The substitution step is particularly important in the context of logarithmic equations, where domain restrictions can lead to extraneous solutions. By carefully substituting and simplifying, we safeguard against the acceptance of invalid solutions and ensure the accuracy of our answer. It is a final checkpoint in our problem-solving journey, providing the assurance that our solution is both mathematically sound and logically consistent.
Verifying the Solution
Simplifying the equation after substitution, we have logā(1/2) + logā3 = logā(3/2). Using the product rule of logarithms on the left side, we get logā(1/2 * 3) = logā(3/2). This simplifies to logā(3/2) = logā(3/2), which is a true statement. This verification step is the culmination of our problem-solving process, the moment of truth where we confirm the validity of our solution. The fact that both sides of the equation are equal demonstrates that x = 1/2 is indeed a solution to the original logarithmic equation. This confirmation is not just a formality; it is a crucial step in ensuring the accuracy and completeness of our answer. It provides the confidence that our solution is not an extraneous root, arising from some algebraic manipulation, but a genuine solution that satisfies the initial conditions of the problem. Verifying the solution is a testament to the rigor of mathematical thinking, where each step is scrutinized and every conclusion is tested against the evidence. In this case, the evidence clearly supports the validity of x = 1/2, confirming its place as the correct answer. This process of verification reinforces the importance of thoroughness in mathematical problem-solving, where a careful check can prevent errors and ensure the integrity of the final result. With the solution verified, we can confidently conclude our journey, knowing that we have successfully navigated the complexities of the logarithmic equation.
Final Answer
Therefore, the solution to the equation logāx + logā3 = logā(x+1) is x = 1/2. This final answer is the culmination of our step-by-step journey through the complexities of the logarithmic equation. We began by understanding the fundamental properties of logarithms, particularly the product rule, which allowed us to simplify the equation. We then equated the arguments, transforming the logarithmic equation into a more manageable algebraic one. Solving for x, we arrived at the potential solution x = 1/2. However, our process did not end there. We recognized the importance of verifying our solution, substituting it back into the original equation to ensure its validity. This crucial step confirmed that x = 1/2 is indeed a solution, satisfying the equation without violating any domain restrictions. This entire process highlights the importance of a systematic approach to problem-solving, where each step is logically connected and rigorously verified. The final answer, x = 1/2, is not just a number; it is the result of a careful and methodical application of mathematical principles. It is a testament to the power of logarithms in simplifying and solving equations, and it represents a successful conclusion to our exploration of this particular logarithmic challenge. With this solution in hand, we can confidently move forward, equipped with a deeper understanding of logarithmic functions and their applications.
Correct Answer: A. x = 1/2
