Solving Logarithmic Equations: A Step-by-Step Guide

Welcome! Let's dive into the world of logarithmic equations and learn how to solve them accurately. This guide will walk you through the process step-by-step, ensuring you understand the core concepts and can confidently tackle these types of problems. We'll pay close attention to the domain of logarithmic functions, a crucial aspect of solving these equations correctly. As a specific example, we'll solve the equation log₃(x-2) = 1, ensuring we find the exact answer and reject any extraneous solutions.

Understanding Logarithmic Equations

First, let's establish a solid foundation. A logarithmic equation is an equation that involves logarithms. Logarithms are the inverse operations of exponentiation. In simpler terms, if we have an equation in the form logₐ(b) = c, it means that a raised to the power of c equals b (aᶜ = b). The 'a' is the base, 'b' is the argument (or the number you're taking the log of), and 'c' is the exponent or the solution to the logarithm. Understanding this relationship is key to solving logarithmic equations. Furthermore, the domain is the set of all possible input values (x-values) for which the logarithm is defined. The argument of a logarithm (the 'b' in logₐ(b)) must always be positive. This constraint is vital, and we'll apply it when solving our example equation. The base, 'a', must be a positive number and cannot be equal to 1. This is because 1 raised to any power will always be 1, so it cannot uniquely identify different values. Keep these principles in mind as we proceed, as they're essential for avoiding common pitfalls and ensuring accurate solutions. We'll use these principles to guide our steps and ensure that our final answers are mathematically valid. Remember, every time we encounter a logarithmic expression, we must think about its domain.

Let’s translate the general logarithmic form logₐ(b) = c into its exponential form. This can be understood as "what power do we need to raise the base ‘a’ to, in order to get ‘b’ as a result?" The answer is the exponent c. We can rewrite the logarithmic equation into its equivalent exponential form, using the base of the logarithm as the base of the exponentiation and the value that the logarithm equals as the exponent. In our example, we are working with base 3 logarithms. We’re asking “to what power do we need to raise 3 to, in order to get x-2 as a result?” That power is 1. This gives us the equation 3¹ = x-2. The exponential form gives us a more direct route to solve for the unknown variable. This is a standard and common approach when solving logarithmic equations, transforming the logarithmic function into a simpler, more manageable form. Doing this makes it easy to find the value of x, our unknown variable. Remember that this transformation will only produce equivalent equations. Remember to always double-check the answers to avoid errors when working on logarithmic equations. This approach emphasizes the inverse relationship between logarithms and exponentiation.

Finally, we will discuss the domain restrictions. The argument of the logarithm, (x-2) in our case, must always be positive. This leads to the inequality x-2 > 0, which means x > 2. Any value of x that is less than or equal to 2 is not in the domain of the original logarithmic expression, and if we encounter such values when solving, we must reject them as extraneous solutions. Extraneous solutions arise when we perform algebraic manipulations that introduce additional solutions that do not satisfy the original equation or the domain restrictions. Therefore, always remember to verify your final answers to avoid these pitfalls, checking the initial domain restrictions. This ensures that you have found valid solutions within the defined range. By keeping these points in mind, we lay the groundwork for a systematic solution process.

Solving log₃(x-2) = 1: Step-by-Step

Now, let's solve the specific equation log₃(x-2) = 1. We will methodically proceed through each stage, applying what we have learned to find the correct answer and to ensure that it satisfies all necessary constraints. This will allow us to cement our understanding of how to approach similar problems. Remember, the goal is not only to find the numerical solution but also to grasp the underlying principles. This is to ensure you can confidently solve any similar problems. We will convert the logarithmic equation to an exponential form, solve for x, and check our answer to ensure it meets the domain restrictions. Let’s start the solving process.

Step 1: Convert to Exponential Form

The first step is to rewrite the logarithmic equation in its equivalent exponential form. Remember the fundamental relationship: logₐ(b) = c is equivalent to aᶜ = b. In our case, the base (a) is 3, the argument (b) is (x-2), and the exponent (c) is 1. Applying this transformation, we get: 3¹ = x - 2.

Step 2: Solve for x

Simplify the exponential equation we obtained in Step 1. 3¹ is simply 3, so our equation becomes: 3 = x - 2. To isolate x, add 2 to both sides of the equation. This yields: x = 3 + 2, therefore, x = 5.

Step 3: Check the Solution

This is a critical step: we must ensure that our solution, x = 5, is valid. The argument of the original logarithm is (x-2). We must check if (x-2) is greater than zero for our solution. Substituting x = 5, we get 5 - 2 = 3. Since 3 > 0, our solution is within the domain of the original logarithmic expression. Thus, x = 5 is a valid solution.

Conclusion and Exact Answer

We successfully solved the logarithmic equation log₃(x-2) = 1. Through a systematic approach, converting to exponential form, solving for x, and verifying the solution against the domain constraints, we found that x = 5. Therefore, the exact answer is x = 5. There are no extraneous solutions in this case. This step-by-step method can be applied to many similar problems. This structured approach helps ensure accuracy and reinforces the importance of understanding the properties of logarithms. This methodical approach will also help you to develop a deeper understanding of the concepts involved and to solve similar problems correctly.

Always double-check your answer and remember to test your result in the original equation to ensure it yields a true statement. Practicing more examples will boost your confidence and proficiency in solving logarithmic equations. Remember, it’s not only about finding the answer but also understanding why the answer is correct and what constraints apply. Through these steps, you'll be well-prepared to tackle a wide variety of logarithmic equations confidently. Mastery comes with practice, so keep practicing to reinforce your understanding. Always remember to check your solutions in the original equation, as this is a crucial step to avoid mistakes.

For further practice and understanding, you can explore more examples on Khan Academy's Logarithm Section.