Simplify Logarithms: Log(x² - X - 2) - Log(x² - 1)

Hey guys! Today, we're diving into the fascinating world of logarithms. We're going to tackle a common problem: expressing a difference of logarithms as a single logarithm and then simplifying it. Specifically, we’ll be working with the expression: log(x² - x - 2) - log(x² - 1). Buckle up, because this is going to be an exciting ride!

Understanding the Core Concepts of Logarithms

Before we jump into the problem, let's refresh our understanding of logarithms. Logarithms are essentially the inverse operation of exponentiation. In simpler terms, if we have an equation like b^y = x, the logarithm (base b) of x is y. This is written as log_b(x) = y. For example, log₁₀(100) = 2 because 10² = 100. When we see "log" without a base specified, it usually means we're dealing with the common logarithm, which has a base of 10. Logarithms have several important properties that we'll use extensively in this problem. One of the most crucial properties is the quotient rule, which states that the logarithm of a quotient is the difference of the logarithms. Mathematically, this is expressed as: log_b(x/y) = log_b(x) - log_b(y). This rule is precisely what we need to combine the two logarithmic terms in our expression into a single logarithm. Another important property is the power rule, which says that the logarithm of a number raised to a power is the product of the power and the logarithm of the number. That is, log_b(x^p) = p * log_b(x). Also, recall the identity property: log_b(b) = 1. These logarithmic identities are our tools, and with practice, we'll become quite adept at using them. In the problem at hand, we're focusing on the quotient rule to consolidate the logarithmic expressions. We will also use our algebra skills, such as factoring quadratic expressions, to simplify the argument of the logarithm once we've combined the terms. Understanding these fundamental logarithmic properties and algebraic techniques is crucial not just for this problem but for various mathematical challenges involving logarithms and exponents. Remember, guys, the more we practice, the more comfortable we become with these concepts.

Applying the Quotient Rule of Logarithms

Now, let's apply the quotient rule to our expression: log(x² - x - 2) - log(x² - 1). According to the quotient rule, we can rewrite this as a single logarithm of a quotient: log((x² - x - 2) / (x² - 1)). This step is a crucial transformation because it combines the two separate logarithmic terms into one. We've essentially condensed the expression, which makes it easier to work with. Now, the focus shifts to simplifying the argument inside the logarithm, which is the fraction (x² - x - 2) / (x² - 1). To simplify this fraction, we need to factor both the numerator and the denominator. Factoring is a fundamental algebraic technique that allows us to break down complex expressions into simpler ones. In this case, we have two quadratic expressions, and factoring them will reveal common factors that can be canceled out. The numerator, x² - x - 2, is a quadratic trinomial. We are looking for two numbers that multiply to -2 and add to -1. Those numbers are -2 and 1. Therefore, we can factor the numerator as (x - 2)(x + 1). Similarly, the denominator, x² - 1, is a difference of squares. We can factor it as (x - 1)(x + 1). Once we have factored both the numerator and the denominator, we can rewrite our logarithmic expression as: log(((x - 2)(x + 1)) / ((x - 1)(x + 1))). Guys, this is where the magic happens! We can now see that there is a common factor of (x + 1) in both the numerator and the denominator. We can cancel out this common factor, which simplifies the fraction even further. This cancellation is valid as long as x ≠ -1, because division by zero is undefined. After canceling the common factor, we are left with the simplified fraction (x - 2) / (x - 1). Thus, our original logarithmic expression is now reduced to log((x - 2) / (x - 1)). This single logarithm represents the simplified form of the original expression, which is exactly what we set out to achieve. By applying the quotient rule and factoring, we've successfully combined and simplified the logarithmic expression.

Simplifying the Argument by Factoring

As we saw in the previous section, applying the quotient rule got us to log((x² - x - 2) / (x² - 1)). Now, the real fun begins – simplifying the argument inside the logarithm. The argument is the fraction (x² - x - 2) / (x² - 1), and our goal is to reduce it to its simplest form. To do this, we need to factor both the numerator and the denominator. Let’s start with the numerator, x² - x - 2. This is a quadratic expression, and we’re looking for two numbers that multiply to -2 (the constant term) and add up to -1 (the coefficient of the x term). Those numbers are -2 and +1. So, we can factor the numerator as (x - 2)(x + 1). Now, let's tackle the denominator, x² - 1. This expression is a classic example of the difference of squares. The difference of squares pattern is a² - b² = (a - b)(a + b). In our case, a is x and b is 1, so we can factor the denominator as (x - 1)(x + 1). Now that we've factored both the numerator and the denominator, we can rewrite the fraction inside the logarithm as (((x - 2)(x + 1)) / ((x - 1)(x + 1))). Guys, can you see the simplification opportunity here? We have a common factor of (x + 1) in both the numerator and the denominator. As long as x is not equal to -1 (because that would make the denominator zero, which is undefined), we can cancel out this common factor. This gives us the simplified fraction (x - 2) / (x - 1). So, our original logarithmic expression log((x² - x - 2) / (x² - 1)) has now been simplified to log((x - 2) / (x - 1)). This is a much cleaner and simpler form. Factoring is such a powerful technique in algebra, and it's essential for simplifying expressions like this. By factoring and canceling common factors, we've made the expression much easier to understand and work with. It’s like taking a complicated puzzle and fitting the pieces together perfectly!

Combining Steps and Final Simplification

Alright, let's recap the steps we've taken so far and bring it all home. We started with the expression log(x² - x - 2) - log(x² - 1). Our mission was to express this as a single logarithm and simplify it as much as possible. First, we applied the quotient rule of logarithms, which allowed us to rewrite the difference of two logarithms as a single logarithm of a quotient. This gave us log((x² - x - 2) / (x² - 1)). Next, we focused on simplifying the fraction inside the logarithm. We factored both the numerator and the denominator. The numerator, x² - x - 2, factored into (x - 2)(x + 1). The denominator, x² - 1, factored into (x - 1)(x + 1). This transformed our expression into log(((x - 2)(x + 1)) / ((x - 1)(x + 1))). We then identified and canceled the common factor of (x + 1) in both the numerator and the denominator, remembering the condition that x ≠ -1 to avoid division by zero. This simplification step resulted in the fraction (x - 2) / (x - 1). Now, we substitute this simplified fraction back into the logarithm, giving us log((x - 2) / (x - 1)). Guys, this is our final simplified expression! We've successfully expressed the original difference of logarithms as a single logarithm and simplified it by factoring and canceling common factors. This final form, log((x - 2) / (x - 1)), is much cleaner and easier to work with than our initial expression. It showcases the power of logarithmic properties and algebraic techniques like factoring. By combining these tools, we can tackle complex expressions and simplify them into more manageable forms. Remember, the key to success in math is understanding the underlying concepts and practicing applying them. We've seen how the quotient rule and factoring can work together to simplify logarithmic expressions. Keep practicing, and you'll become a pro at this in no time!

Conclusion

So, to wrap things up, we've taken the expression log(x² - x - 2) - log(x² - 1), and through the magic of logarithmic properties and factoring, we've simplified it to log((x - 2) / (x - 1)). We started by applying the quotient rule to combine the two logarithms into one. Then, we factored the quadratic expressions in the numerator and denominator, which allowed us to cancel out a common factor. This process highlights the importance of understanding both logarithmic identities and algebraic techniques. These skills are not only useful for simplifying expressions but also for solving equations and understanding more advanced mathematical concepts. Guys, remember that mathematics is like building with LEGOs – each concept builds upon the previous one. Mastering these foundational skills will make you a more confident and capable mathematician. Keep practicing, keep exploring, and most importantly, keep having fun with math! We've successfully navigated this problem, and I hope you feel more confident in your ability to tackle similar challenges. Until next time, keep those logarithmic and algebraic gears turning!