Solve Logarithmic Equation: Step-by-Step Guide

Aug 12, 2025 by Pedro Alvarez 47 views

Solving Logarithmic Equations: A Comprehensive Guide

Hey everyone! Today, we're diving deep into the fascinating world of logarithmic equations. You know, those equations with logs that sometimes look intimidating? Well, don't worry! We're going to break it down step by step, making sure you not only understand the mechanics but also the why behind each step. We'll tackle a specific problem as an example, but the principles we cover will be applicable to a wide range of logarithmic equations. So, grab your thinking caps, and let's get started!

Understanding Logarithms: The Basics

Before we jump into solving, let's quickly recap what logarithms are all about. Think of a logarithm as the inverse operation of exponentiation. In simpler terms, if we have an equation like 2^3 = 8, the logarithm answers the question: "To what power must we raise 2 to get 8?" The answer, of course, is 3. We write this logarithmically as log₂ 8 = 3. Here, 2 is the base of the logarithm, 8 is the argument, and 3 is the result.

Key Components: It's crucial to understand these key components: the base (the number being raised to a power), the argument (the number we want to obtain), and the exponent (the power to which we raise the base). The logarithmic form, logb a = c, is equivalent to the exponential form, b^c = a. Getting comfortable with converting between these forms is a fundamental step in mastering logarithmic equations.
Logarithmic Properties: To effectively solve logarithmic equations, we need to have a firm grasp on some key logarithmic properties. These properties act as our tools, allowing us to manipulate and simplify equations. The product rule states that logb (mn) = logb m + logb n. The quotient rule says that logb (m/n) = logb m - logb n. And the power rule tells us that logb (mp) = p logb m. These rules might seem abstract now, but we'll see them in action soon, making them much clearer. Remember, these properties are derived from the properties of exponents, so understanding that connection can make them easier to remember.
Common Logarithms and Natural Logarithms: Two special types of logarithms deserve a mention: common logarithms and natural logarithms. A common logarithm is a logarithm with a base of 10, written as log₁₀ or simply log. A natural logarithm has a base of e (Euler's number, approximately 2.71828), written as ln. Most calculators have dedicated buttons for these logarithms, making them easy to work with. Understanding these special cases is crucial because they often appear in various applications, especially in science and engineering.

The Problem: log₂ (x-1)² - log₂ (x²-4x+3) = log₂ 4

Now, let's dive into the specific equation we're going to solve: log₂ (x-1)² - log₂ (x²-4x+3) = log₂ 4. This equation might look a bit daunting at first glance, but don't worry, we'll break it down systematically. The key here is to use the logarithmic properties we just discussed to simplify the equation and isolate the variable x. We'll also need to be mindful of the domain of logarithmic functions, as we can't take the logarithm of a negative number or zero.

Initial Assessment: Before we start manipulating the equation, let's take a moment to assess it. We see that all the logarithms have the same base (2), which is excellent news because it means we can use the quotient rule to combine the terms on the left-hand side. We also need to keep in mind that the arguments of the logarithms, (x-1)² and (x²-4x+3), must be greater than zero. This will help us determine the valid solutions later on. A careful initial assessment can save us from making mistakes down the line.
Strategy: Our strategy for solving this equation will involve several key steps. First, we'll use the quotient rule of logarithms to combine the two logarithms on the left-hand side into a single logarithm. This will simplify the equation significantly. Next, we'll use the property that if logb m = logb n, then m = n. This allows us to eliminate the logarithms and obtain an algebraic equation. Finally, we'll solve the algebraic equation and check our solutions against the domain restrictions to ensure they are valid. This step-by-step approach will help us stay organized and avoid errors.
Potential Pitfalls: It's important to be aware of potential pitfalls when solving logarithmic equations. One common mistake is forgetting to check the solutions against the domain restrictions. Another is misapplying the logarithmic properties. It's also crucial to be careful with algebraic manipulations, especially when dealing with squared terms. By being mindful of these potential pitfalls, we can increase our chances of finding the correct solutions. Keep an eye out for these common errors as we work through the problem.

Step-by-Step Solution

Alright, let's get our hands dirty and solve this equation step-by-step. We'll take each step carefully, explaining the reasoning behind it. This isn't just about getting the answer; it's about understanding the process.

Applying the Quotient Rule: The first step is to use the quotient rule of logarithms, which states that logb m - logb n = logb (m/n). Applying this to our equation, we get: log₂ [(x-1)² / (x²-4x+3)] = log₂ 4. This simplifies the left-hand side into a single logarithm, making the equation much easier to work with. Notice how the quotient rule allows us to combine the two logarithmic terms into one, effectively reducing the complexity of the equation.
Eliminating Logarithms: Now that we have a single logarithm on each side of the equation, we can use the property that if logb m = logb n, then m = n. This means we can eliminate the logarithms and set the arguments equal to each other: (x-1)² / (x²-4x+3) = 4. This is a crucial step because it transforms the logarithmic equation into an algebraic equation, which we can solve using familiar techniques. This property is a direct consequence of the fact that the logarithmic function is one-to-one.
Simplifying the Algebraic Equation: Let's simplify the algebraic equation. First, we can factor the denominator: x² - 4x + 3 = (x-1)(x-3). So, our equation becomes (x-1)² / [(x-1)(x-3)] = 4. We can further simplify by canceling the common factor of (x-1) in the numerator and denominator, but we need to be cautious here. Canceling a factor means we are assuming that factor is not equal to zero. So, we need to remember that x ≠ 1. After canceling, we have (x-1) / (x-3) = 4. Simplifying algebraic expressions is a crucial skill in solving logarithmic equations, as it often leads to a more manageable equation.
Solving for x: Now we have a simple rational equation. Let's solve for x. Multiply both sides by (x-3) to get rid of the fraction: x-1 = 4(x-3). Expanding the right-hand side, we have x-1 = 4x - 12. Now, let's isolate x. Subtract x from both sides: -1 = 3x - 12. Add 12 to both sides: 11 = 3x. Finally, divide by 3: x = 11/3. So, we have a potential solution, but we need to check if it's valid.

Checking for Extraneous Solutions

This is a super important step! Remember, we can't take the logarithm of a negative number or zero. So, we need to make sure our solution, x = 11/3, doesn't make any of the arguments in the original equation negative or zero. This step is crucial because sometimes we might find solutions that don't actually work in the original equation – these are called extraneous solutions. Always, always, always check your solutions in the original equation!

Domain Restrictions: First, let's identify the domain restrictions. For log₂ (x-1)², we need (x-1)² > 0. This means x-1 ≠ 0, so x ≠ 1. For log₂ (x²-4x+3), we need x²-4x+3 > 0. We already factored this as (x-1)(x-3), so we need (x-1)(x-3) > 0. This inequality holds when x < 1 or x > 3. Combining these restrictions, we have x < 1 or x > 3, and x ≠ 1. Understanding and determining the domain restrictions is a fundamental step in solving logarithmic equations, as it helps us identify potential extraneous solutions.
Verification: Now, let's check our solution, x = 11/3. First, 11/3 ≠ 1, so that condition is satisfied. Next, 11/3 is greater than 3, so it also satisfies the condition x > 3. Therefore, x = 11/3 is a valid solution. If our solution had violated any of the domain restrictions, we would have had to discard it. The verification process ensures that our solution is not extraneous and actually satisfies the original equation.
Extraneous Solutions Explained: Extraneous solutions can arise when we perform operations that are not reversible, such as squaring both sides of an equation or, in this case, simplifying logarithmic expressions. These operations can introduce solutions that don't satisfy the original equation. That's why checking our solutions against the domain restrictions is so important. Understanding the concept of extraneous solutions is crucial for solving various types of equations, including logarithmic, radical, and rational equations.

Final Answer

After all that work, we've arrived at our final answer! The solution to the equation log₂ (x-1)² - log₂ (x²-4x+3) = log₂ 4 is x = 11/3.

Recap of the Solution Process: Let's quickly recap the steps we took to solve this equation. First, we applied the quotient rule of logarithms to combine the logarithmic terms. Then, we eliminated the logarithms by setting the arguments equal to each other. We simplified the resulting algebraic equation by factoring and canceling common factors. We solved the simplified equation for x. Finally, and most importantly, we checked our solution against the domain restrictions to ensure it was valid. This systematic approach is a powerful tool for solving logarithmic equations.
Importance of Showing Your Work: It's important to show all your work when solving mathematical problems, especially logarithmic equations. This not only helps you keep track of your steps and avoid errors but also allows others to follow your reasoning. Showing your work also makes it easier to identify mistakes if you make them. Clear and organized work is a hallmark of a strong mathematical thinker.
Practice Makes Perfect: Solving logarithmic equations can be challenging at first, but with practice, it becomes much easier. The more problems you solve, the more comfortable you'll become with the logarithmic properties and the solution process. So, don't be afraid to tackle a variety of logarithmic equations. Practice is the key to mastering any mathematical skill.

Tips and Tricks for Solving Logarithmic Equations

Let's wrap things up with some handy tips and tricks that can make solving logarithmic equations even smoother. These are the little nuggets of wisdom that come from experience, and they can really help you out in the long run.

Simplify First: Before you do anything else, try to simplify the equation as much as possible. This might involve using logarithmic properties to combine terms, factoring algebraic expressions, or canceling common factors. A simplified equation is almost always easier to solve. Look for opportunities to simplify the equation early on, as this can save you time and effort in the long run.
Isolate the Logarithm: If possible, isolate the logarithmic term on one side of the equation. This makes it easier to eliminate the logarithm by using the property that logb m = logb n implies m = n, or by converting the equation to exponential form. Isolating the logarithmic term is a crucial step in many logarithmic equations.
Check for Extraneous Solutions (Again!): We can't stress this enough! Always check your solutions against the domain restrictions. This is the single most important step in solving logarithmic equations, as it prevents you from accepting incorrect answers. Make checking for extraneous solutions a habit, not an afterthought.
Convert to Exponential Form: If you're stuck, try converting the logarithmic equation to its equivalent exponential form. This can sometimes make the equation easier to solve, especially if the logarithm is isolated. Remember the relationship between logarithmic and exponential forms: logb a = c is equivalent to b^c = a.
Use Substitution: For more complex equations, try using substitution. If you see a repeated expression, like (logb x)², try substituting y = logb x. This can transform the equation into a more familiar form, such as a quadratic equation. Substitution is a powerful technique for simplifying complex equations.

So there you have it, guys! A comprehensive guide to solving logarithmic equations. We've covered the basics, worked through an example problem step-by-step, and shared some valuable tips and tricks. Now it's your turn to put these skills into practice. Remember, the key is to understand the underlying principles and to approach each problem systematically. Happy solving!