Single Logarithm: Simplify 3 Log₂x - (log₂3 - Log₂(x+4))

Hey guys! Today, we're diving into the fascinating world of logarithms, and we're going to tackle a problem that might seem a bit daunting at first. Our mission, should we choose to accept it, is to express the logarithmic expression $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$ as a single, unified logarithm. Sounds like fun, right? Let's break it down step by step and make sure everyone's on board.

Understanding Logarithmic Properties

Before we jump into the problem, let's quickly recap some key logarithmic properties that we'll be using. These properties are like the secret sauce that makes simplifying logarithmic expressions possible. Trust me, once you've got these down, you'll feel like a log wizard!

1. Power Rule: This is a big one. The power rule states that $a \log_b c = \log_b (c^a)$. In simpler terms, if you have a number multiplying a logarithm, you can bring that number inside as an exponent. For example, $2 \log_3 5$ can be rewritten as $\log_3 (5^2)$, which is $\log_3 25$. See how we took that 2 and made it an exponent? This will be super handy for our problem.

2. Quotient Rule: Another crucial property. The quotient rule says that $\log_b a - \log_b c = \log_b (\frac{a}{c})$. This means if you're subtracting two logarithms with the same base, you can combine them into a single logarithm by dividing the arguments. Think of it as subtraction turning into division inside the log. For instance, $\log_2 10 - \log_2 5$ becomes $\log_2 (\frac{10}{5})$, which simplifies to $\log_2 2$.

3. Product Rule: Last but not least, the product rule states that $\log_b a + \log_b c = \log_b (ac)$. Similar to the quotient rule, but for addition. If you're adding two logarithms with the same base, you can combine them into a single logarithm by multiplying the arguments. So, $\log_5 3 + \log_5 4$ transforms into $\log_5 (3 \times 4)$, which equals $\log_5 12$.

These three properties – the power rule, the quotient rule, and the product rule – are the building blocks we need to simplify and combine logarithms. They might seem abstract now, but as we work through our problem, you'll see how they come to life and make complex expressions much more manageable.

Step-by-Step Simplification

Okay, with our logarithmic toolkit ready, let's tackle the expression: $3 \log _2 x-\left(\log _2 3-\log _2(x+4)\right)$. Our goal is to condense this into a single logarithm, and we'll do it systematically, making sure we don't miss any steps.

Step 1: Applying the Power Rule

The first part of our expression is $3 \log _2 x$. Remember the power rule? We can bring that 3 up as an exponent of $x$. So, $3 \log _2 x$ becomes $\log _2 (x^3)$. This is a great start – we've already simplified one part of the expression.

Step 2: Distributing the Negative Sign

Next, we need to deal with the parentheses: $-\left(\log _2 3-\log _2(x+4)\right)$. Let's distribute that negative sign to both terms inside the parentheses. This gives us $-\log _2 3 + \log _2(x+4)$. It’s crucial to handle the negative sign carefully; it’s a common place where mistakes can happen.

Now our expression looks like this: $\log _2 (x^3) - \log _2 3 + \log _2(x+4)$. We're getting closer to our goal of a single logarithm!

Step 3: Applying the Quotient Rule

We have a subtraction between two logarithms: $\log _2 (x^3) - \log _2 3$. This is where the quotient rule comes into play. We can combine these two logarithms into a single logarithm by dividing their arguments. So, $\log _2 (x^3) - \log _2 3$ becomes $\log _2 \left(\frac{x^3}{3}\right)$.

Our expression is now: $\log _2 \left(\frac{x^3}{3}\right) + \log _2(x+4)$. Notice how we're systematically reducing the number of logarithmic terms.

Step 4: Applying the Product Rule

Finally, we have an addition of two logarithms: $\log _2 \left(\frac{x^3}{3}\right) + \log _2(x+4)$. The product rule is perfect for this situation. We can combine these logarithms by multiplying their arguments. This means we multiply $\frac{x^3}{3}$ by $(x+4)$.

So, $\log _2 \left(\frac{x^3}{3}\right) + \log _2(x+4)$ becomes $\log _2 \left(\frac{x^3(x+4)}{3}\right)$.

The Final Simplified Expression

We did it! After applying the power rule, distributing the negative sign, using the quotient rule, and finally employing the product rule, we've successfully expressed the original expression as a single logarithm:

$\log _2 \left(\frac{x^3(x+4)}{3}\right)$

This is our final answer. It looks much cleaner and more concise than the original expression, right? The key was to take it one step at a time, applying the logarithmic properties in the correct order.

Common Mistakes to Avoid

Logarithmic expressions can be tricky, and there are a few common pitfalls to watch out for. Knowing these mistakes can save you a lot of headaches.

1. Incorrectly Applying the Power Rule: A frequent error is forgetting to apply the power rule before other operations. Remember, if you have a coefficient multiplying a logarithm, bring it inside as an exponent first.

2. Mixing Up the Quotient and Product Rules: It's easy to get the quotient and product rules mixed up. Just remember: subtraction inside logarithms corresponds to division, and addition corresponds to multiplication. Thinking of it that way can help you keep them straight.

3. Forgetting the Base: Always pay attention to the base of the logarithm. You can only combine logarithms using the quotient and product rules if they have the same base. If the bases are different, you'll need to use other techniques, like the change of base formula.

4. Distributing Negatives Incorrectly: As we saw in our example, distributing a negative sign can be tricky. Make sure you apply the negative to every term inside the parentheses.

5. Skipping Steps: It might be tempting to rush through the simplification process, but skipping steps can lead to errors. Take your time and write out each step clearly.

By being aware of these common mistakes, you can approach logarithmic expressions with confidence and minimize the chances of making errors. Practice makes perfect, so the more you work with these types of problems, the better you'll become at avoiding these pitfalls.

Practice Problems

To really solidify your understanding, let's try a few more practice problems. Working through these will help you get comfortable with the logarithmic properties and the simplification process.

1. Simplify: $2 \log_3 x + \log_3 5 - \log_3 (x-1)$

2. Express as a single logarithm: $\log_4 (x+2) + \log_4 (x-2) - 3 \log_4 x$

3. Combine into a single logarithm: $\frac{1}{2} \log_5 9 + 2 \log_5 x - \log_5 3$

Try working through these problems on your own, and then check your answers. Remember to use the power rule, quotient rule, and product rule in the correct order. Don't be afraid to make mistakes – that's how we learn!

Conclusion

So there you have it! We've successfully transformed a complex logarithmic expression into a single logarithm using the power rule, quotient rule, and product rule. We've also discussed common mistakes to avoid and worked through some practice problems. Logarithms might seem intimidating at first, but with a solid understanding of the properties and a bit of practice, you can conquer them with confidence. Keep practicing, and you'll become a log whiz in no time! Remember, the power rule helps with exponents, the quotient rule simplifies subtraction, and the product rule tackles addition. Happy logging, guys!