Solving Logarithmic Equations Log 2(x+21)=3 An In-Depth Guide

Logarithmic equations can seem daunting at first, but with a systematic approach and a solid understanding of logarithmic properties, they become quite manageable. In this comprehensive guide, we will delve into the process of solving the logarithmic equation log₂(x + 21) = 3. We will meticulously break down each step, ensuring clarity and fostering a deep understanding. We will also address the crucial aspect of verifying solutions, particularly the need to reject extraneous solutions that do not fall within the domain of the original logarithmic expression. This is a critical step in solving logarithmic equations, as logarithmic functions are only defined for positive arguments. By the end of this guide, you will be well-equipped to tackle similar logarithmic equations with confidence.

To begin, let's restate the equation we aim to solve: log₂(x + 21) = 3. This equation involves a logarithm with base 2. The expression inside the logarithm, (x + 21), is known as the argument. The equation essentially asks: "To what power must we raise 2 to obtain (x + 21)?" The answer, according to the equation, is 3. The fundamental property we will employ to solve this equation is the definition of a logarithm. The logarithmic equation logₐ(b) = c is equivalent to the exponential equation aᶜ = b. This property allows us to convert the logarithmic equation into its equivalent exponential form, which is often easier to solve. Applying this property to our equation, log₂(x + 21) = 3, we can rewrite it as 2³ = x + 21. This transformation is the cornerstone of solving logarithmic equations. By converting the equation to exponential form, we eliminate the logarithm, making it easier to isolate the variable x. This step is crucial and forms the basis for the subsequent steps in the solution process. In the following sections, we will continue with the solution process, solve for x, and then meticulously verify the solution to ensure it is valid.

Now, let’s simplify the exponential equation 2³ = x + 21. We know that 2³ equals 2 * 2 * 2, which is 8. So, we can rewrite the equation as 8 = x + 21. This simplification is a straightforward arithmetic operation, but it's an essential step in isolating the variable x. We now have a simple linear equation, which is much easier to solve than the original logarithmic equation. To isolate x, we need to subtract 21 from both sides of the equation. This maintains the equality and moves us closer to finding the value of x. Subtracting 21 from both sides gives us 8 - 21 = x + 21 - 21, which simplifies to -13 = x. Therefore, we have found a potential solution: x = -13. However, we must remember that this is just a potential solution. Before we can confidently declare it as the solution, we need to verify it by plugging it back into the original logarithmic equation and checking if it satisfies the equation and if it falls within the domain of the logarithmic function. This verification step is paramount in solving logarithmic equations, as it helps us identify and eliminate any extraneous solutions that may arise due to the nature of logarithmic functions. In the next section, we will meticulously perform this verification step to ensure the validity of our solution.

Verifying the solution is a critical step when dealing with logarithmic equations. Logarithmic functions have a restricted domain; specifically, the argument of a logarithm must be strictly positive. In other words, we can only take the logarithm of positive numbers. If we plug a non-positive number (zero or a negative number) into the argument of a logarithm, the expression is undefined. This is a fundamental property of logarithmic functions and must be considered when solving logarithmic equations. Our original equation was log₂(x + 21) = 3, and we found a potential solution of x = -13. To verify this solution, we must substitute x = -13 back into the original equation and check if the equation holds true and if the argument of the logarithm remains positive. Let's substitute x = -13 into the argument (x + 21). We get (-13) + 21 = 8. Since 8 is a positive number, the argument of the logarithm is positive when x = -13. This is the first condition for the solution to be valid. Now, let's substitute x = -13 into the original equation: log₂((-13) + 21) = log₂(8). We know that 2³ = 8, so log₂(8) = 3. Therefore, the equation log₂((-13) + 21) = 3 holds true. Since x = -13 satisfies both the condition that the argument of the logarithm is positive and the original equation, it is indeed a valid solution. This careful verification process underscores the importance of checking solutions in logarithmic equations. By verifying the solution, we ensure that it is not an extraneous solution and that it truly satisfies the given equation.

Therefore, after diligently solving the equation and meticulously verifying the solution, we can confidently state that the solution to the logarithmic equation log₂(x + 21) = 3 is x = -13. This solution satisfies the original equation and lies within the domain of the logarithmic function. The process we followed exemplifies the standard approach to solving logarithmic equations, which involves converting the logarithmic equation to its equivalent exponential form, solving for the variable, and then, crucially, verifying the solution. This verification step is essential to avoid extraneous solutions, which can arise due to the domain restrictions of logarithmic functions. By understanding and applying these steps, you can confidently solve a wide range of logarithmic equations. The key takeaways from this solution process are the importance of the definition of a logarithm, the conversion between logarithmic and exponential forms, and the necessity of verifying solutions. These concepts are fundamental to working with logarithmic functions and solving logarithmic equations.

In summary, solving logarithmic equations requires a systematic approach. The key steps involve converting the logarithmic equation to its equivalent exponential form, solving for the unknown variable, and, most importantly, verifying the solution by ensuring it satisfies the original equation and lies within the domain of the logarithmic function. The equation log₂(x + 21) = 3, which we have solved in this guide, serves as a prime example of this process. By following these steps meticulously, you can confidently tackle a variety of logarithmic equations. Remember that the domain of a logarithmic function is restricted to positive arguments, and this restriction plays a crucial role in verifying solutions. This comprehensive guide has equipped you with the knowledge and skills to solve logarithmic equations effectively. By mastering these techniques, you will enhance your understanding of logarithmic functions and their applications in various mathematical and scientific contexts. Continue practicing with different types of logarithmic equations to further solidify your skills and build confidence in your problem-solving abilities.