Express $3 \log _3(w-3)-4 \log _3(w+8)$ As A Single Logarithm

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Hey guys! Today, we're diving into the exciting world of logarithms! Logarithms might seem a bit intimidating at first, but trust me, they're super useful and quite fun once you get the hang of them. In this article, we're going to tackle a common type of problem: expressing a logarithmic expression as a single logarithm. This is a fundamental skill in algebra and calculus, and mastering it will make your life much easier when dealing with logarithmic equations and functions. So, let's jump right in and break down the process step by step.

Understanding Logarithms

Before we dive into the specifics of combining logarithmic expressions, let's quickly recap what logarithms are all about. A logarithm is essentially the inverse operation of exponentiation. Think of it this way: if you have an exponential equation like by=xb^y = x, the logarithm answers the question, "What exponent do I need to raise b to, in order to get x?" This is written as log⁡b(x)=y\log_b(x) = y. Here, b is the base of the logarithm, x is the argument, and y is the exponent. Understanding this basic relationship is crucial for manipulating logarithmic expressions effectively. Remember, guys, the logarithm is just another way of asking what power we need to raise the base to get a certain number. It's like a secret code that unlocks the exponent! We'll be using this understanding as we move forward, so make sure you're comfortable with the core concept.

Basic Logarithmic Properties

To express multiple logarithmic terms as a single logarithm, we need to use the properties of logarithms. These properties act as our toolkit, allowing us to manipulate logarithmic expressions in a structured way. There are three key properties that we'll be using extensively:

  1. Product Rule: log⁡b(mn)=log⁡b(m)+log⁡b(n)\log_b(mn) = \log_b(m) + \log_b(n). This rule tells us that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Think of it as splitting a product inside the logarithm into separate logarithmic terms added together.
  2. Quotient Rule: log⁡b(mn)=log⁡b(m)−log⁡b(n)\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n). This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. It's the counterpart to the product rule, dealing with division instead of multiplication.
  3. Power Rule: log⁡b(mp)=plog⁡b(m)\log_b(m^p) = p \log_b(m). This rule is super powerful (pun intended!). It allows us to bring exponents inside the logarithm out as coefficients, and vice versa. This is especially useful when we have terms like 3log⁡3(w−3)3 \log_3(w-3) in our original expression. Knowing these rules is like having the keys to unlock and transform logarithmic expressions. Remember, the goal is to use these properties to compress multiple terms into a single, neat logarithm.

Problem Statement

Alright, let's get to the heart of the matter. We're given the expression: $3 \log _3(w-3)-4 \log _3(w+8)$ Our mission, should we choose to accept it (and we do!), is to rewrite this expression as a single logarithm. This means we want to combine these two logarithmic terms into one, using the properties we just discussed. To do this effectively, we need to carefully apply the logarithmic properties in the correct order. This might seem a bit like detective work, where we're piecing together clues to solve the puzzle. But don't worry, we'll walk through each step together, and by the end, you'll be a pro at this. The key is to break down the problem into smaller, manageable steps, and that's exactly what we're going to do.

Step-by-Step Solution

Step 1: Apply the Power Rule

Our first order of business is to deal with the coefficients in front of the logarithms. Remember the power rule? It says that log⁡b(mp)=plog⁡b(m)\log_b(m^p) = p \log_b(m). We can use this rule in reverse to bring those coefficients inside the logarithms as exponents. So, let's apply the power rule to both terms in our expression:

  • 3log⁡3(w−3)3 \log _3(w-3) becomes log⁡3((w−3)3)\log _3((w-3)^3)
  • 4log⁡3(w+8)4 \log _3(w+8) becomes log⁡3((w+8)4)\log _3((w+8)^4)

Now our expression looks like this: $\log _3((w-3)^3) - \log _3((w+8)^4)$ See how the coefficients have disappeared and become exponents? This is a crucial first step in combining the logarithms. By using the power rule, we've set the stage for applying the quotient rule next.

Step 2: Apply the Quotient Rule

Now that we've taken care of the coefficients, we can focus on combining the two logarithmic terms. Notice that we have a subtraction between the two logarithms. This is our cue to use the quotient rule, which states that log⁡b(mn)=log⁡b(m)−log⁡b(n)\log_b(\frac{m}{n}) = \log_b(m) - \log_b(n). We can apply this rule in reverse to combine our two logarithms into one. Think of the first logarithmic term as mm and the second as nn. Applying the quotient rule, we get:

log⁡3((w−3)3)−log⁡3((w+8)4)=log⁡3((w−3)3(w+8)4)\log _3((w-3)^3) - \log _3((w+8)^4) = \log _3(\frac{(w-3)^3}{(w+8)^4})

Voilà! We've successfully combined the two logarithms into a single logarithm. The subtraction has transformed into a division inside the logarithm. This is the magic of the quotient rule at work. Notice how the structure of the expression has changed from two separate terms to a single, more compact form. This is exactly what we were aiming for.

Final Answer

After applying the power rule and then the quotient rule, we've successfully expressed the original expression as a single logarithm. The final answer is: $\log _3(\frac{(w-3)3}{(w+8)4})$ This is the simplified form of the expression, where all the logarithmic terms have been combined into one. Remember, guys, the key to solving these problems is to carefully apply the properties of logarithms in the correct order. Each step builds on the previous one, leading us closer to the final solution. By breaking down the problem into smaller steps, we can tackle even the most complex logarithmic expressions with confidence.

Conclusion

Expressing logarithmic expressions as a single logarithm is a valuable skill in mathematics. By understanding and applying the properties of logarithms, we can simplify complex expressions and make them easier to work with. In this article, we took a step-by-step approach to solving the problem, emphasizing the power rule and the quotient rule. Remember, the key is to practice and familiarize yourself with these properties. The more you work with them, the more natural they will become. Keep practicing, guys, and you'll become logarithm masters in no time!

This skill is not just about simplifying expressions; it's also crucial for solving logarithmic equations, understanding exponential growth and decay, and even in fields like computer science and engineering. So, the effort you put into mastering these concepts will pay off in the long run. And remember, don't be afraid to ask questions and seek help when you need it. Learning mathematics is a journey, and we're all in it together. So keep exploring, keep learning, and keep having fun with math!