Solving Logarithmic Equations A Step-by-Step Guide

In the realm of mathematics, solving logarithmic equations can sometimes seem like navigating a complex maze. However, by following a structured approach and understanding the fundamental properties of logarithms, these equations can be tackled with confidence. This article delves into a detailed, step-by-step solution of the logarithmic equation log₃(x+2) = log₃(2x²-1). We aim to provide a clear and comprehensive guide, ensuring that each step is meticulously explained. This article serves as a valuable resource for students, educators, and anyone seeking to enhance their problem-solving skills in mathematics. Understanding logarithms is crucial in various fields, including engineering, computer science, and finance, making this a fundamental concept in mathematical literacy. The key to successfully navigating logarithmic equations lies in a systematic approach, and we are here to guide you through each twist and turn of the process. Mastering this type of equation not only boosts your mathematical prowess but also enhances your analytical thinking abilities, applicable in numerous real-world scenarios. We'll break down each step, ensuring you grasp not just the "how" but also the "why" behind each manipulation. Logarithmic equations may initially appear daunting, but with a solid understanding of the underlying principles, you'll find them both manageable and even fascinating. So, let's embark on this mathematical journey together, demystifying logarithmic equations one step at a time.

Step 1: Equating the Arguments

When faced with a logarithmic equation where the bases are the same, the first crucial step is to equate the arguments. This stems from the fundamental property of logarithms: if logₐ(b) = logₐ(c), then b = c, provided that 'a' is a positive number not equal to 1, and 'b' and 'c' are positive. In our equation, log₃(x+2) = log₃(2x²-1), the base is 3, which satisfies the condition. Therefore, we can confidently equate the arguments (x+2) and (2x²-1), leading us to the equation x + 2 = 2x² - 1. This step is pivotal as it transforms the logarithmic equation into a more familiar algebraic form, specifically a quadratic equation. The beauty of this transformation lies in its ability to simplify the problem, allowing us to leverage our existing knowledge of algebraic techniques to find the solution. However, it is crucial to remember that this step is valid only when the logarithmic bases are identical. Equating arguments effectively eliminates the logarithmic function, paving the way for solving for the unknown variable, 'x' in this case. It's a powerful technique that streamlines the solving process, making it more accessible and less intimidating. This initial step is the cornerstone of solving logarithmic equations, setting the stage for the subsequent algebraic manipulations that will lead us to the solution. Remember, the key is to isolate the logarithmic terms and then apply this fundamental property to transform the equation into a solvable form.

Step 2: Rearranging into a Quadratic Equation

Following the equation of arguments, the next essential step is to rearrange the equation into a standard quadratic form. This form, ax² + bx + c = 0, is crucial because it allows us to apply various techniques such as factoring, completing the square, or using the quadratic formula to find the solutions. Starting with our equation x + 2 = 2x² - 1, we aim to bring all terms to one side, setting the equation equal to zero. This involves subtracting x and 2 from both sides of the equation. This process yields 0 = 2x² - x - 3, which is now a quadratic equation in the standard form. The transformation into standard form is not just a matter of aesthetics; it's a strategic move that unlocks a toolkit of methods for finding solutions. Recognizing the quadratic form is a key skill in algebra, and it's essential for solving a wide range of mathematical problems, not just logarithmic equations. The coefficients a, b, and c in the quadratic equation (in this case, 2, -1, and -3, respectively) play a crucial role in determining the solutions. Being able to identify these coefficients correctly is vital for applying the appropriate solution methods. Rearranging into a quadratic equation is a fundamental algebraic technique, and its application here demonstrates the interconnectedness of different mathematical concepts. This step bridges the gap between logarithmic and algebraic problem-solving, highlighting the importance of a strong foundation in both areas. With the equation now in standard quadratic form, we are well-positioned to move on to the next phase: finding the values of x that satisfy the equation.

Step 3: Factoring the Quadratic Equation

Once the quadratic equation is in the standard form, the next step is to factor the quadratic expression. Factoring is a technique that involves breaking down the quadratic expression into a product of two binomials. This method, when applicable, provides a direct route to finding the solutions (also known as roots) of the equation. Our quadratic equation is 0 = 2x² - x - 3. To factor this expression, we look for two binomials of the form (Ax + B)(Cx + D) such that when multiplied, they yield the original quadratic expression. The key is to find two numbers that multiply to give the product of the leading coefficient (2) and the constant term (-3), which is -6, and add up to the middle coefficient (-1). These numbers are -3 and 2. Using these numbers, we can rewrite the middle term of the quadratic expression and then factor by grouping. This process leads us to the factored form (2x - 3)(x + 1) = 0. Factoring is not always straightforward, and it requires practice and a good understanding of number properties. However, when successful, it offers a concise and elegant way to solve quadratic equations. The factored form immediately reveals the values of x that make the expression equal to zero, as each factor represents a potential solution. If factoring proves difficult, alternative methods such as the quadratic formula can be employed. However, mastering factoring is a valuable skill in algebra, and it often leads to quicker solutions than other methods. By successfully factoring the quadratic equation, we have essentially transformed a complex expression into a simpler form, making it easier to identify the roots of the equation. This step is a testament to the power of algebraic manipulation in solving mathematical problems. The factored form of the equation sets the stage for the next logical step: identifying the potential solutions.

Step 4: Identifying Potential Solutions

After factoring the quadratic equation, the next crucial step is to identify the potential solutions. This involves setting each factor equal to zero and solving for x. Our factored equation is (2x - 3)(x + 1) = 0. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This principle allows us to split the equation into two simpler equations: 2x - 3 = 0 and x + 1 = 0. Solving 2x - 3 = 0 for x involves adding 3 to both sides and then dividing by 2, which gives us x = 3/2. Solving x + 1 = 0 for x involves subtracting 1 from both sides, which gives us x = -1. Therefore, the potential solutions to our quadratic equation are 3/2 and -1. It's important to emphasize that these are potential solutions and not necessarily the solutions to the original logarithmic equation. This is because the domain of logarithmic functions is restricted to positive arguments. The process of identifying potential solutions is a direct application of the zero-product property, a fundamental concept in algebra. It's a powerful technique that allows us to transform a factored equation into a set of simpler equations, each of which can be solved independently. However, it's crucial to remember that these solutions are only potential until they are verified against the original equation. This verification step is essential to ensure that the solutions are valid and not extraneous. By identifying the potential solutions, we have narrowed down the possibilities and are one step closer to solving the original logarithmic equation. The next step involves a critical check to ensure the validity of these solutions within the context of the logarithmic functions involved.

Step 5: Checking for Extraneous Solutions

The pivotal step in solving logarithmic equations is checking for extraneous solutions. This is a critical process because not all potential solutions obtained from solving the transformed algebraic equation will satisfy the original logarithmic equation. Extraneous solutions arise due to the restricted domain of logarithmic functions; the argument of a logarithm must always be positive. Our potential solutions are x = 3/2 and x = -1. To check these, we must substitute each value back into the original equation, log₃(x+2) = log₃(2x²-1), and verify that the arguments of the logarithms remain positive. Let's start with x = 3/2. Substituting this value into the equation, we get log₃(3/2 + 2) = log₃(7/2) and log₃(2(3/2)² - 1) = log₃(2(9/4) - 1) = log₃(9/2 - 1) = log₃(7/2). Since both arguments are positive (7/2 > 0), x = 3/2 is a valid solution. Now, let's check x = -1. Substituting this value into the equation, we get log₃(-1 + 2) = log₃(1) and log₃(2(-1)² - 1) = log₃(2 - 1) = log₃(1). While both sides of the equation are equal, we must also check that the arguments are positive. In this case, both arguments are 1, which is positive, so x = -1 is also a valid solution. However, if substituting a potential solution resulted in a negative or zero argument for any logarithm, that solution would be deemed extraneous and discarded. This step highlights the importance of understanding the domain restrictions of logarithmic functions. Checking for extraneous solutions is not just a formality; it's a necessary step to ensure the accuracy and validity of the final answer. By meticulously verifying each potential solution, we can confidently identify the true solutions to the logarithmic equation. This careful approach distinguishes a correct solution from a potential pitfall, ensuring a robust understanding of logarithmic equations.

Step 6: Stating the Valid Solutions

Having meticulously checked for extraneous solutions, the final step is to state the valid solutions to the original logarithmic equation. This step is crucial as it solidifies the answer and provides a clear conclusion to the problem-solving process. From our previous steps, we identified x = 3/2 and x = -1 as potential solutions to the equation log₃(x+2) = log₃(2x²-1). After substituting these values back into the original equation and verifying that the arguments of the logarithms remained positive, we confirmed that both potential solutions are indeed valid. Therefore, the solutions to the equation log₃(x+2) = log₃(2x²-1) are x = 3/2 and x = -1. Stating the solutions clearly and explicitly is not just a matter of completing the problem; it also demonstrates a thorough understanding of the solution process. It showcases that we have not only found potential solutions but also rigorously verified their validity within the context of the original equation. This step provides closure to the problem and ensures that the answer is presented in a clear and unambiguous manner. It's important to note that in some cases, logarithmic equations may have no solutions, one solution, or multiple solutions. The process of checking for extraneous solutions is what ultimately determines the final set of valid solutions. By clearly stating the valid solutions, we communicate the outcome of our mathematical journey in a concise and understandable way. This final step is the culmination of all the previous steps, bringing the problem-solving process to a satisfying conclusion.

Conclusion

In conclusion, solving logarithmic equations requires a systematic approach that combines algebraic manipulation with a thorough understanding of logarithmic properties. We have demonstrated a step-by-step method for solving the equation log₃(x+2) = log₃(2x²-1), which includes equating arguments, rearranging into a quadratic equation, factoring, identifying potential solutions, checking for extraneous solutions, and finally, stating the valid solutions. The key takeaways from this process are the importance of the domain restrictions of logarithmic functions and the necessity of checking for extraneous solutions. By following these steps carefully, you can confidently tackle a wide range of logarithmic equations. Mastering logarithmic equations is not only a valuable skill in mathematics but also enhances problem-solving abilities applicable in various fields. The ability to break down complex problems into manageable steps and to apply the correct techniques is a hallmark of mathematical proficiency. This detailed guide serves as a valuable resource for anyone seeking to improve their understanding of logarithmic equations and their solution methods. By practicing these techniques and understanding the underlying principles, you can develop a strong foundation in mathematics and enhance your analytical thinking skills. Logarithmic equations, while sometimes challenging, can be approached with confidence and solved effectively with the right knowledge and approach. Remember, the journey through mathematics is one of continuous learning and refinement, and each problem solved is a step forward in expanding your mathematical horizons.