Solving Logarithmic Equations Find The Value Of $^2\log 48 - ^2\log 3$

Hey guys! Ever stumbled upon a logarithmic equation and felt a bit lost? No worries, we've all been there. Logarithms might seem intimidating at first, but they're actually pretty cool tools for simplifying complex calculations. In this article, we're going to break down a specific problem: finding the value of $^2\log 48 - ^2\log 3$. We'll go through the steps together, making sure everything is crystal clear. So, buckle up and let's dive into the world of logarithms!

Understanding Logarithms: The Basics

Before we jump into the problem, let's quickly recap what logarithms are all about. Think of a logarithm as the inverse operation of exponentiation. In simple terms, if we have an equation like $b^x = y$, the logarithm asks the question: "To what power must we raise b to get y?" This is written as $\log_b y = x$.

• Base: The b in $\log_b y$ is called the base of the logarithm. It's the number that's being raised to a power.
• Argument: The y is the argument of the logarithm – it's the number we want to find the logarithm of.
• Result: The x is the result – it's the power to which we must raise b to get y.

For example, $\log_2 8 = 3$ because $2^3 = 8$. Here, the base is 2, the argument is 8, and the result is 3. Now that we've refreshed the basics, let's tackle some key logarithmic properties that will help us solve our problem.

Key Logarithmic Properties

Logarithms have several properties that make them incredibly useful for simplifying expressions. The two properties we'll use in this problem are:

1. Quotient Rule: $\log_b (x/y) = \log_b x - \log_b y$ This rule states that the logarithm of a quotient is equal to the difference of the logarithms.
2. Power Rule: $\log_b (x^p) = p \log_b x$ This rule tells us that the logarithm of a number raised to a power is equal to the power times the logarithm of the number.

These properties are like our secret weapons in the world of logarithms. With these tools in our arsenal, we can simplify and solve complex logarithmic expressions. Let's keep these in mind as we move forward and apply them to our main problem.

Applying Logarithmic Properties to Solve $^2\log 48 - ^2\log 3$

Okay, let's get back to our problem: finding the value of $^2\log 48 - ^2\log 3$. Notice that both logarithms have the same base, which is 2. This is excellent news because it means we can use the quotient rule to simplify the expression. Remember, the quotient rule states that $\log_b (x/y) = \log_b x - \log_b y$.

In our case, $b = 2$, $x = 48$, and $y = 3$. Applying the quotient rule, we get:

$^2\log 48 - ^2\log 3 = ^2\log (48/3)$

Now, we just need to simplify the fraction inside the logarithm:

$^2\log (48/3) = ^2\log 16$

Great! We've simplified the expression to $^2\log 16$. Now, the question is: "To what power must we raise 2 to get 16?" Let's think about the powers of 2:

• $2^1 = 2$
• $2^2 = 4$
• $2^3 = 8$
• $2^4 = 16$

Bingo! We see that $2^4 = 16$. Therefore, $^2\log 16 = 4$. So, the value of $^2\log 48 - ^2\log 3$ is 4. Awesome, right? We've successfully navigated through the problem using logarithmic properties. Now, let’s delve a bit deeper and explore some alternative methods to tackle this type of problem.

Alternative Methods for Solving Logarithmic Equations

While using the quotient rule is a straightforward method, there are other ways we could have approached this problem. Sometimes, having different tools in your toolbox can make solving problems easier or give you a fresh perspective. One such method involves prime factorization.

Prime Factorization Method

Prime factorization is the process of breaking down a number into its prime factors. Prime numbers are numbers that have only two factors: 1 and themselves (e.g., 2, 3, 5, 7, etc.). Let's apply this method to our problem.

We start by breaking down 48 into its prime factors:

$48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3 = 2^4 × 3$

So, we can rewrite $^2\log 48$ as $^2\log (2^4 × 3)$. Now, we can use another logarithmic property, the product rule, which states that $\log_b (xy) = \log_b x + \log_b y$. Applying this rule, we get:

$^2\log (2^4 × 3) = ^2\log (2^4) + ^2\log 3$

Using the power rule, $\log_b (x^p) = p \log_b x$, we can simplify $^2\log (2^4)$:

$^2\log (2^4) = 4 × ^2\log 2$

Since $^2\log 2 = 1$ (because $2^1 = 2$), we have:

$4 × ^2\log 2 = 4 × 1 = 4$

Now, our original expression $^2\log 48 - ^2\log 3$ becomes:

$(4 + ^2\log 3) - ^2\log 3$

The $^2\log 3$ terms cancel each other out, leaving us with:

$4$

Just like before, we arrive at the same answer: 4. Isn't it cool how different paths can lead to the same destination? Understanding multiple methods can make you a more versatile problem solver. Now, let’s reinforce our understanding with a quick summary and some practice tips.

Summary and Practice Tips for Mastering Logarithms

Alright, guys, let's recap what we've learned today. We tackled the problem of finding the value of $^2\log 48 - ^2\log 3$ and successfully solved it using logarithmic properties. We started with a quick review of the basics of logarithms, including the definition and key terms like base and argument. Then, we applied the quotient rule to simplify the expression and found that $^2\log 48 - ^2\log 3 = ^2\log 16 = 4$.

We also explored an alternative method using prime factorization and the product rule, which further solidified our understanding of logarithmic properties. Remember, the key properties we used were:

• Quotient Rule: $\log_b (x/y) = \log_b x - \log_b y$
• Power Rule: $\log_b (x^p) = p \log_b x$
• Product Rule: $\log_b (xy) = \log_b x + \log_b y$

Now, how can you become a logarithm master? Here are a few tips:

1. Practice Regularly: The more you practice, the more comfortable you'll become with logarithms. Try solving a variety of problems, from simple to complex.
2. Understand the Properties: Make sure you truly understand the logarithmic properties. Don't just memorize them – know how and why they work.
3. Break Down Problems: When faced with a complex problem, break it down into smaller, more manageable steps. This will make the problem less intimidating and easier to solve.
4. Check Your Work: Always double-check your work to ensure you haven't made any mistakes. Logarithms can be tricky, so it's good to be thorough.
5. Use Resources: There are tons of resources available online and in textbooks that can help you learn more about logarithms. Don't be afraid to explore and use them.

By following these tips and practicing regularly, you'll be well on your way to mastering logarithms. Remember, every problem you solve is a step forward in your learning journey. So, keep practicing, keep exploring, and most importantly, have fun with it! Happy solving, guys!